Understanding the field structure for A_5 and more

1. Aug 11, 2007

bham10246

Question 1: Given any degree 5 polynomial f over the rationals, what are the conditions on the roots of f if I want its Galois group to be isomorphic to the alternating group $A_5$?

I know that if f is an irreducible degree 5 polynomial with 3 real roots and 2 complex roots, then its Galois group contains a 2-cycle and a 5-cycle, which implies Gal(f) is isomorphic to $S_5$. But what about the conditions on the roots in order for the Galois group to be $A_5$?

Question 2: If f is an irreducible degree 5 polynomial whose Galois group is solvable, then are the following true?
Case 1. If f has only two complex roots and three real roots, then Gal(f) is isomorphic to $\mathbb{Z}_5 (semidirect\: product)\: \mathbb{Z}_2$.
Case 2. If f has four complex roots, then Gal(f) has a 5-cycle and two transpositions. So Gal(f) is isomorphic to $\mathbb{Z}_5 (semidirect\: product)\: \mathbb{Z}_2\times \mathbb{Z}_2$?

Question 3: But if the Galois group is abelian and f is an irreducible polynomial of degree 5 with at least 2 complex roots, isn't the field extension corresponding to this Galois group isomorphic to a subgroup of the cyclotomic extension? Thus if the polynomial is irreducible of degree 5 with at least 2 complex roots, then could Gal(f) be isomorphic to $\mathbb{Z}_5$? I think that it cannot, but I cannot come up with a valid argument.

Thank you.

2. Aug 12, 2007

matt grime

Doesn't the discriminant do what you want? If it is a square then it's A_5, otherwise it is S_5? Something like that?

3. Aug 12, 2007

matt grime

As Chris Hilmann pointed out to me (private correspondence), the discriminant only tells you whether the Galois group is a subgroup of A_5 or S_5.

4. Aug 12, 2007

bham10246

Hi Matt grime, you're probably right. I'll look through Wikipedia.

5. Aug 12, 2007

mathwonk

ramblings

the galois group of a polynomial over Q say, permutes the roots in such a way that certain combinations of them remain unchanged. in particular the coefficients of the polynomial remain unchanged since they are rational, i.e. the product of the roots (the constant term) remains unchanged, their sum (+ or - the second highest term) remains unchanged,...

now the product of all pairwise differences also remains unchanged, but every pair of roots occurs twice in this product as c-d and again as d-c.

so if we only include one difference for each pair, we get a number that changes sign under any transposition, and hence only remains unchanged by even permutations.

so if we can see whetehr the galois group leaves that number alone, we will know whether the galois group consists on even opermutations. now the first more inclusive product is invariant under all permutations, hence also under the full galois group, hence is a rational number, called the discriminant.

moreover there is a (in general very complicated) formula for that number in terms of the coefficients of the original polynomial, e.g. b^2 - 4c, for a quadratic.

now the crucial number we want to see about is the square root of this discriminant. so we get that our galois group is contained ni the even group of permutations, i,.e. A(n), if and only if the square root of the discriminant is preserved, if and only if the square root of the discriminant lies in the abse field Q.

in studying irreducible cubics, this suffices to determine the galois group.

for quiartics one seeks other semi symmetric combinations of the roots, and tests to see if they are preserved by the galois group. this gives a lengthy list of tests for various known subgroups of S(4) to be the galois group, as given in hungerford.

more practical tests, involve looking for elements of the galois group which are cycles of various lengths, by reducing the polynomial mod various primes, as given in van der waerden. even though one may not determine the group prciely, one does find certain elements that may be of interest.

geometric versions of these tests are used in algebraic geometry to find the galois groups of various extensions of function fields.

e.g. every dominant map X-->Y of degree n, between irreducible algebraic varieties of the same dimension, determines a finite field extension of degree n between their function fields. the galois group of the normal closure of this extension is sometimes computable as the monodromy group of the map.

i.e. even though most fibers of the map have n elements, in a "branch" fiber some elements come together into a fiber with fewer elements. this produces a branching behavior upstairs as one goes around loops which encircle the branch value downstairs.

i.e. in any covering space, the fundamental group at a point downstairs acts on the fiber over that point. by removing branch points we obtain a covering space, and fixing any good point gives a representation of the fundamental group on the fiber over that point.

e.g. the squaring map C-->C taking z to z^2, is branched only over 0, so the covering map of C-0 has a monodromy rZpresentation of the fundamental group Z of the downstairs copy of C-0 into the permutation groip of the fiber (1,-1} say over 1. this representation has image Z/2Z, the galois group of the inclusion of functions fields C(Z^2) in C(Z).

[i.e. the function field C(Z) of the bae pulls back by this map to the subfield C(Z^2) of the field upstairs.]

thus for a degree n map we represent the fundamental group of any general point downstairs, in the permutation group S(n). the image of this map is the galois group of the field extension of the map. it can hopefully computed from the geometric branching behavior of the map

e.g. the map of degre 27 taking a pair (S,L) consisting of a cubic surface S in 3 space and a line L on the that surface to S, had galois group equal to the famous group E6 or E8 or something, of order about 51,840.

It dos not equal S(27) because when lines are permuted, any two that meet on the cubic surface must go to two others that meet, so the "incidence relation" of lines is preserved both by the monodromy group and hence by the galois group.

Interestingly, for reasons arising from the interplay between cubic surfaces, cubic threefolds, conic bundles, and their connection to prym varieties, the prym map onto abelian varieties of dimension 5 has degree 27 and its monodromy group is also this group.

wirtinger stated without proof in 1895 that this prym map had finite degree, which Beauvile proved in 1976, then i made progress on the question of computing the degree in my phd thesis in 1977, introducing ideas to compute degrees by examining infinite fibers. the monodromy group was computed by Donagi about 1979, when he discovered a beautiful "tetragonal" relation in the fibers of the prym map that mirrored the incidence relation of lines on a cubic surface, but in terms of linear series on curves.

some of this story (donagi's tetragonal relation) is described in the chapter on prym varieties in the book "complex abelian varieties", by birkenhake and lange, which chapter is an expansion of lectures i gave in erlangen.

Last edited: Aug 12, 2007
6. Aug 13, 2007

mathwonk

to make a connection to the thread on discriminants, the branch locus in the geometric case, i.e. the points downstairs over which there are fewer points in the fibers, is called the discriminant locus of the (finite) map.

E.g,. in the case of a polynomial of a given degree, like a quadratic, if we consider the downstairs space Y as the space of all quadratics Q of form X^2+bX+c, with coordinates b,c, and the upstairs space X as the set of all pairs, (Q,p) where p is a root of Q, then the map sending (Q,p) to Q is a map of degree 2 which is branched exactly over the discriminant locus of those quadratics where b^2-4c = 0, since these are the ones with a double root.

so the usual discriminant of a polynomial defines the discriminant locus of this special map. In general, if we have a map X-->Y, the discriminant locus downstairs may be thought of as those points over which the fiber is singular, i.e. not a manifold. (a doubled point is not a zero dimensional manifold, but two simple points are.)

this locus is called N(0) in the case of principally polarized abelian varieties and their theta divisors, and was the key to Mumford's proof that A(g) has general type for g >6. Beauville's result on A(4) was that 4 dimensional jacobians form exactly one component of N(0). It had been shown earlier by Mayer that Jacobians were one component, in his thesis. Varley and I later showed the discriminant locus in A(5) also has two components and gave a shorter proof of Beauville's result for A(4). Then Debarre generalized the result for all g.

In the case of the Prym map above onto A(5), Donagi showed that the branch locus of the Prym map was exactly one component of the branch locus N(0). Varley and I then showed that component was also the locus of intermediate jacobians of quartic double solids. we then wrote some papers showing how to compute the tangent cone to a discriminant locus by taking a vcertain projective mapping of the singular locus itself of the fiber over a given point, and taking its projective dual variety in a certain cycle theoretic sense.

Last edited: Aug 13, 2007
7. Aug 13, 2007

mathwonk

this geometric version also has an algebraic version, oin th galois theory of ring extensions. i.e. if one has an embedding of domains R-->S with S finite over R, (i.e. integral), they induce an algebraic embedding of fields ff(R)-->ff(S), and the galois theory of this field extension is mirrored in the gometry of the map of affine schemes spec(S)-->spec(R), i.e. pulling back prime ideals from S to R.

the prime ideals should be thiought of as points and the number of ideals over a given ideal can drop below the maximum number juist as wih geometric maops of points, and the galois group acts by permuting these ideals over a given ideal. i have never mastered this theory, but it fascinates me, and can be used to discuss the specialization of galois groups mod various primes, as in Lang's Algebra book.

8. Aug 13, 2007

mathwonk

here is a famous example. consider the space of all cubic surfaces in P^3. and over this space consider the space of those surfaces. what i mean is downstairs there is one point for each surface, and upstairs there is a cubic surface over each point.

i.e. Y = space of all cubic polynomials on P^3, and X = all pairs (f.x) where f is a cubic polyonnmial and x is a root of f.

thus X-->Y is a map of relative dimension 2, such that over each point of Y there is a cubic surface.

the discriminant locus of this map is the locus in Y of cubic polynomials defining a singular surface.

now consider the related map Z-->Y such over each point of Y representing a cubic surface, the fiber is the set of lines on that cubic surface. it is classical that this map has degree 27, and that both these maps have the same discriminant locus. i.e. a cubic surface is singular if and only if it haS FEwER THAN 27 lines on it.

then it was surprizing when herb Clemens noted, based on a remark of Beauville, that for quintic threefolds this is not true, i.e. the generic quintic threefold has a finite number of lines, maybe 2875, but the braNCH LOCUS OF LINES IS DIFFERENT FROM THAT of the threefolds, and this extends to other rational curves. (I may have this wrong.) But this allowed Clemens to show that the Griffiths group of a quintic threefold is not finitely generated, a major result.

I.e. I am guessing (based on old memories) that the fact that curves of different degrees had different branch loci allowed a proof that they are inependent in the GRIFFITHS GROUP.

Last edited: Aug 13, 2007
9. Aug 14, 2007

Chris Hillman

Pedantic correction and some suggested (non)-reading

First of all, I'm not really here at PF, I just happened to stumble over this thread while Googling.

Second, what I actually PM'd to Matt Grime is the same as what mathwonk said a few hours later: the discriminant tells you whether or not the Galois group contains any odd permutations, i.e. whether or not it is a subgroup of the alternating group. (I know Matt knows this, he must have been tired, since what he wrote is a bit garbled.)

Third,

Bham, noooo!

(Covers eyes in horror.)

At one point in 2006, I was one of the 500 most active Wikipedians
http://en.wikipedia.org/wiki/User:Hillman/Archive
http://en.wikipedia.org/w/index.php...Wikipedians_by_number_of_edits&oldid=84344861
but I gradually and reluctantly concluded that Wikipedia has not only failed in its stated mission of providing a free, on-line, comprehensive and reliable encyclopedia, it has succeeded in redefining "good information" from "accurate information" to "easily obtained and easily manipulated information". You seem like a serious student, bham, so I hope you will appreciate without being told that inaccuracies and misdirections are particularly harmful for students of the exact sciences. I won't try to explain here my disenchantment with the Wikipedia, but I'll say this: because of my very extensive experience at Wikipedia (roughly August 2005-August 2006) my views are not easily dismissed as being based upon ignorance of Wikipedia, which is indeed a world unto itself, which plays by its own (perennially controversial, wildly inconsistent, and largely unwritten) rules.

The point which is relevant here is that there is a widely believed myth to the effect that "mathematics articles in Wikipedia are generally reliable" even though physics articles have been bedeviled by crankery, vandalism, and so forth. Unfortunately, the truth is that math is not immune the same kinds of problems experienced in other regions of the Wikipedia. While at present you can find good stuff in Wikipedia, the bottom line is that you should never use Wikipedia unless you plan to check everything you read against other sources (which in my view defeats the purpose of an encyclopedia). In other words, in my view, only knowledgeable experts can use Wikipedia more or less "safely", since they are more likely to sense deception or misinformation in a given article.

Aside to the moderators: Wikipedia continues to have many passionate fans, some of whom are active at PF, so please ensure that this thread not be hijacked into an attack on my criticism of Wikipedia, since what should interest posters here is the large body of interesting mathematics related to Galois groups. I suggest that anyone who wishes to attack my criticism of the Wikipedia, which I have sketched in a previous PF thread in another sub-forum, should start a new thread in the General Discussion subforum. (But since I am not really here at PF, don't expect me to join in!)

Bham, in preference to the Wikipedia, let me recommend some sources of information which are more stable and, I would argue, far more reliable:

In addition to Hungerford, some books which discuss computing Galois groups systematically are:

D. J. H. Garling, A Course in Galois Theory, Cambridge University Press, 1986.

B. L. van der Waerden, Algebra, Springer, 1991, two volumes (reprint of classic graduate textbook, often called the first "modern" algebra text)

A book which doesn't discuss general algorithms but which many students find useful is

Lisl Gaal, Classical Galois Theory with Examples, Chelsea, Fourth Ed., 1998.

Two excellent historically oriented books are:

B. L. van der Waerden, A History of Algebra : from al-Khwarizmi to Emmy Noether, Springer, 1985.

Jean-Pierre Tignol, Galois's Theory of Algebraic Equations, World Scientific, 2001.

(At least one of the two book by van der Waerden does give a general algorithm related to invariants like the discriminant, along the lines discussed by mathwonk, but I can't recall which one.)

For background on invariants of finite groups (the discriminant suggests correctly that if one can find some NON-invariants of the unknown Galois group, this can provide useful information about this group), I recommend chapter two of

Bernd Sturmfels, Algorithms in Invariant Theory, Springer, 1993.

You will be interested in "Deck groups" in homotopy theory. (If you don't yet know what these are, you'll understand why you are interested in them after you learn what they are!) I recommend

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002

(see in particular the discussion of how generators and relations, i.e. describing the Deck groups as quotients of a free group, correspond to the topology of "coverings", and note the Galois duality between coverings and Deck groups--- the figure is something I drew when I was an undergraduate, BTW. Hatcher's book was in the works for decades before it finally appeared and is also available over the InterNet; I think it is worthwhile to buy the printed version but maybe that is just me).

An excellent if less lavishly illustrated book discussing Deck groups is:

W. S. Massey, Algebraic Topology: an Introduction, Springer, 1967

(which only covers homotopy theory, but does that very well indeed).

There are many other directions you can go with this. Permutation groups and group actions are a lovely topic in their own right. Generators and relations (as in covering maps and Deck groups) might lead you to the lovely notion of reflection groups and the ADE paradigm. And lurking in the background here is one of "the most important topics in mathematics which hardly anyone studies", Kleinian geometry. Some good books for exploring these might be:

For group actions generally:

Peter M. Neumann et al., Groups and Geometry, Oxford University Press, 1999.

For permutation groups generally:

Peter M. Cameron, Permutation Groups, Cambridge University Press, 1999.

(Note in particular how permutation groups can be used to solve problems in enumerative combinatorics, the science of answering questions like: "how many trees with n-vertices exist, up to graph-theoretic isomorphism?"

For reflection groups:

Richard M. Kane, Reflection Groups and Invariant Theory, Springer, 2001

For group actions in algebraic geometry and some hints about Kleinian geometry:

Joe Harris, Algebraic Geometry: a First Course, Springer, 1992.

A fascinating topic in algebraic geometry barely hinted at in this book is Schubert calculus, which is a way of answering questions like: "given a generic arrangement of eight 3-dimensional flats in five dimensional complex projective space, how many lines meet all eight?" (Fourteen, if I have not goofed.) It turns out that Schubert calculus is in turn closely related to many other fascinating topics, including reflection groups and parts of invariant theory and representation theory. Schubert calculus was notoriously controversial in the nineteenth century, and was not put on a sound footing until much later, and there is much interest in finding a rigorous but easier and more elegant foundation.

You should know that invariant theory meets algebraic geometry meets symmetry in "geometric invariant theory". All of these topics might be said to have originated in the nineteenth century in the theory of algebraic curves, which is a fascinating topic in its own right. See

C. Herbert Clemens, A scrapbook of complex curve theory, Plenum Press, 1980

for an overview (same Clemens mentioned by mathwonk), and see

C. G. Gibson, Elementary Geometry of Algebraic Curves, Cambridge University Press, 1998

for a systematic and very readable introduction.

All of this recommended reading is merely preparation for this bit of intelligence: John Baez of Weekly fame http://www.math.ucr.edu/home/baez/TWF.html has told me that he intends in future Weeks to explain a "categorification" of Schubert calculus.

You might have heard of "Kleinian groups"; this is related to Kleinian geometry (and reflection groups), but the basic ideas of Kleinian geometry are far more elementary.

So what is Kleinian geometry? Basically, a circle of ideas establishing an equivalence between group actions and "geometries". Klein and his followers had much to say about how to find the group action associated with a given "geometry"; later Tits and his followers had much to say about how to find the "geometry" associated with a given group action. You can play this game both for "continuum geometries" (such as the real projective plane, or the ordinary sphere) and "discrete geometries" (such as finite projective spaces).

The "continuum geometries" might be a bit easier to grapple with. One recent and very readable book which briefly discusses some basic ideas of Kleinian geometry from the point of view of the theory of homogeneous spaces in differentiable manifolds is

Chris J. Isham, Modern Differential Geometry for Physicists, World Scientific, 2005.

In one of John's recent posts, he discusses the idea of "invariant relations"; incidence relations of the kind which are the subject of Schubert calculus are examples of this. In particular, the symmetry group of the 27 lines on a cubic surface, which was mentioned by mathwonk, shows how enforcing incidence relations can restrict a symmetry group to be smaller than it would be otherwise.

Returning to the original problem of actually computing Galois groups, let me add a generalization: maximally efficient computation of Galois groups usually seems to involve some "trick" or insight particular to a given situtation, but there are of course systematic approaches for small degree, in fact Maple can compute Galois groups polynomials of small degree, at least over the rationals, and other sofware such as GAP can do much more. (J. H. Conway is one of the people who are very good at computing Galois groups, BTW. IIRC, his work is used in the Maple algorithm.)

If you play around with this, you'll soon find that most polynomials have maximally symmetric and hence boring roots. More interesting are the ones with smaller Galois groups, since this indicates that in an algebraic sense, the disposition of the roots is "asymmetrical". This suggests an "inverse problem": find polynomials with a given Galois group. This is a challenging problem in itself, one which has a growing literature! In fact, the question of which groups arise as Galois groups is a famous open problem.

Lastly, I should probably say that many of these topics are of current interest in theoretical physics. As someone once quipped, "theoretical physics is locally isomorphic to mathematics". To mention just one "teaser" along these lines, the Lorentz group arises as a Galois group! See

Jones and Singerman, Complex Functions, Cambridge University Press, 1987.

It might be worth mentioning that entropy, in various senses both mathematical and physical, is related to symmetry. The connection is elementary and arises from the Galois duality between the (complete) lattice of stabilizer subgroups and the lattice fixsets, for a given group action. Then the stabilizer lattice modulo conjugacy gives a lattice of complexions and conditional complexions which correspond to entropy and conditional entropy in Shannon's theory, or in Boltzmann and Planck's (somewhat naive) early work on statistical mechanics. In this situation, when one has a mapping between two sets acted on by the same group G which respects the actions (often called an "equivariant map" or G-hom), the conditional complexion of a target point given its preimage is always a group, in fact, a generalized Galois group. An interesting problem here is the restoration problem: suppose we have a group action by a finite group equipped with a "special" generating set, and given a scrambled state we wish to find a "short word" unscrambling this state, by some means more efficient than trial and error (see Conway's work on the Rubik cube). Here, the stabfix lattice gives useful information, and the reconstruction involves a kind of cohomology of group actions, which is due to Frobenius.

(Mathwonk mentioned how, when studying equivariant mappings, it can happen that the size of the preimage is less than expected at some point, e.g. a "branch point". The generalized Galois groups can vary from point to point even when the size does not, e.g. in some cellular automatons.)

If you've heard of Noether's theorem in physics, well, that also is related to symmetry and to invariant theory. See

Peter J. Olver, Equivalence, Invariants and Symmetry, Cambridge University Press, 1995

Peter J. Olver, Applications of Lie Groups to Differential Equations, Springer, 2000

Sigh... this long list of suggested reading (I could easily have gone one at even greater length) must seem daunting, and I regret that. It might explain why John has had some difficulty staying on topic in recent weeks--- there are so many fascinating and important side alleys to explore! If nothing else, perhaps this post might inspire some ambitious young person to set out to write the missing multivolume monograph on Kleinian geometry, the subject which ties together all of these topics and many more.

Last but not least, mathwonk, please say more about the symmetries of the 27 lines on a generic cubic surface! And what about the 28 bitangents to a generic quartic plane curve? Klein himself said something about these topics, but clearly much more can be said about the relation between the incidence properties of these geometrical configurations and the algebraic properties of their symmetry groups.

Last edited: Aug 14, 2007
10. Aug 15, 2007

mathwonk

take a cubic surface in P^3 and project it from a point on the surface down to P^3. I believe this is a 2:1 map with branch locus a quartic, and the 27 lines of the cubic surface go down to bitangent lines of that quartic, as does the point of projection, which should blow up to another bitangent line.

anyone who understands this please explain it to me.

11. Aug 15, 2007

matt grime

It is an open question as to what groups are Galois groups of extensions of Q - every group is the galois group of some extension of some field (like every group is the fundamental group of some object).

12. Aug 15, 2007

phoenixthoth

This is like attending a colloquium. I feel like if I can glean 5% of what's being said, that's something. *sighs*

Please excuse me if this is a derailment... I have a couple of questions...

What is Galois for other than determining solvability by radicals to certain polynomial equations? (Yes, I'm quite ignorant)

Is it differential Galois theory in which they prove certain elementary functions don't have elementary antiderivatives? That seems to me pretty intense because I find myself quite unable to tell the difference between these two:

$$\sum\limits_{n=0}^{\infty }\frac{x^{n}}{n!}$$

$$\sum\limits_{n=0}^{\infty }\frac{x^{2n+1}}{\left( 2n+1\right) n!}$$

The last one being the antiderivative of e^(x^2), right?

13. Aug 15, 2007

Chris Hillman

Correction

Thanks, Matt, that was my brain blip, so now we're even

14. Aug 15, 2007

mathwonk

my posts were intended to eveal a connection between galois theory and the atructure of all geometric mappings.

the second topic is quite interesting and easily searchable online. associated to the names of liouville, and more rcntly ritt, rosenlicht, and risch.

the absic theorem says that given any field of functions closed under differentiation, the only elements of that field which have integrals which can be obtained from elementary operations on elements of that field,

are linear combinations of elements of form u'/u, and v', where the u's and v's are in the original field. i.e. the only possible elementary integrals of elements of the given field, are liner combiantions of elements in the field and their logs.

(this uses complex numbers to avoid considering trig functions separately from logs and exponentials.

in specific acses one can show the only possible elementary integral of a functon of form f.e^g, where f is rational, has form R.e^g, where R is rational, and also R' + Rg' = f. so e.g. proving e^(x^2) has no elementary integral reduc es to showing that R' + 2xR = 1, has no rational solution R.

I do not understand the way differential galois theory enters here, but Risch does strengthen Liouvilles statement to one involving an assertion about the linear combination of the functions u'/u, v', being invariant under automorphisms of the extension field in which the solution lies.

see maxwell rosenlicht, Amer Math Monthly, 1972, page 968.

15. Aug 15, 2007

mathwonk

my web algebra notes, both grad versions, contain proofs that every finite abelian group is a galois group over Q, based on a special case of dirichlet's theorem on primes in arithmetic progression.

16. Aug 15, 2007

matt grime

I get the impression, though I am quite ignorant myself, that the Langland's project would be a case to examine.

The point of Galois theory is, in some sense, the study of 'nice' extensions of one field over the other. Or more importantly, it is the study of actions of groups on field extensions, and, in current thinking, understanding such morphisms of objects is the way to understand the objects (the slogan of category theory in some sense).

17. Aug 15, 2007

mathwonk

check out the grothendieck conjecture, that the arithmetic fundamental group of a hyoerbolic curve actually determines the curve.

i.e. given a variety over a field k with algebraic closure K, there is a notion of etale fundamental group and an embedding of the group of X/K (thegeometric fundamental group of X), into that of X/k (the arithmetic fund grp), as a

normal subgroup. the quotient group is the group of the map spec(K)-->spec(k) = Gal(k), the absolute galois group of the extension k in K.

Grothendieck conjectured that for "anabelian" varieties this extension of a fundamental group by the absolute galois group determines X. there has been a lot of work on this for over 20 years now. google it and see.

18. Aug 18, 2007

Chris Hillman

Galois: what did he do and why do we still care?

I'll interpret that to mean: "what ideas did Galois introduce during his brief career in mathematics and what are they good for?"

That's a good question and I reget that I don't have time to offer a truly appropriate answer.

Just a few thoughts off the top of my head:

Galois more or less introduced groups, lattices (in sense of Mac Lane), rings, ideals and fields, which form the core of modern algebra. Along the way he more or less introduced the concept of object, morphism, and subobject, although these were not formalized until the last century with the introduction of category theory. I probably don't need to try to explain why modern algebra and category theory are important in modern mathematics!

In the nineteenth century, "geometry" pretty much meant "algebraic geometry", the geometry of curves and surfaces defined by sets of polynomial equations, especially in the setting of finite dimensional projective spaces over the complex field. Galois more or less introduced something completely novel, finite fields and projective planes over such fields, which are "geometries" with finitely many "points" and "lines". These turn out to have unexpected applications in such areas as algebraic coding theory, but the importance of his innovation goes far beyond finite projective planes themeselves. Indeed, the importance of finite projective planes wasn't recognized until the middle of the last century, when Fano rediscovered these geometries.

More generally, the idea of finding "geometries" hiding in "combinatorial relations" respecting some group action is essentially one half of Klein's Erlangen program, which offers one of the most influential (if least known) organizing principles in all of mathematics. To repeat what I said last time: every "geometrical configuration" has an algebraic doppelganger,an invariant (perhaps an invariant polynomial or an invariant relation) of a suitable group action, and conversely every invariant under a group action (in fact, every groupoid) expresses in algebraic terms some "geometrical relationship" among appropriate "geometric elements" (such as points, lines, and planes).

Turning to Galois theory specifically, as you no doubt know, in his famous 1831 paper, Galois completely solving the ancient problem of expressing the roots of polynomials in terms of radicals (or IOW reducing the problem of expressing the roots of a general polynomial to the roots of a monomial, which we assume we know how to accomplish--- as you probably know, Galois showed this can be done exactly in case the Galois group of the polynomial is solvable. No mean achievement, but sans doubt the most important aspect of this work was that Galois provided an inspiring example of one of the greatest ideas of mathematics: transform a mathematical problem expressed in terms of one theory into the language of another theory, solve it there, and transform the answer back into the original language. This is one of the truly great ideas in intellectual history (around the same time, Gauss was using it in what we now call differential geometry, but this wasn't published until around 1850).

Specifically, Galois more or less introduced what is now called a Galois connection in lattice theory, an "order reversing bijection" between the lattice of "intermediate fields" of a field extension (a bit of anachronism--- these concepts are due to Artin and Noether around 1900) and the lattice of subgroups of the Galois group. The latter are much easier to find, exploiting the poweful and convenient theory of finite groups! Actually, this is the "almost trivial" part of Galois theory, but never underestimate a triviality; it turns out that Galois connections are everywhere in mathematics (and computer science). To mention just one, there is a Galois duality between algebraic varieties (curves and surfaces defined by sets of polynomial equations, the fundamental object of study in "classical" algebraic geometry) and ideals of a polynomial ring (the fundamental object of study in "classical" commutative algebra). See again mathwonk's posts in this thread for many references to this setting!

Well, I should mention one other example of a Galois duality: the one introduced by Galois himself, in which given any action by a group G on a set X, the lattice of pointwise stabilizer subgroups is dual to the lattice of fixsets. Here, the fixset of a subgroup $H < G$ is the set of points, $\triangleright H$ which is fixed by every element of the subgroup H; the stabilizer of a subset A of X is the set of elements of G, $\triangleleft A$, which fix every point in A. The fixset of the stabilizer of A, or $\triangleright \triangleleft A$, is a subset including A, but often much larger, sometimes called the Galois closure of A--- not to be confused with similar terminology in field theory! This can be characterized as the smallest set B such that given knowledge of where an unknown element g of G moves the points of A, you can deduce how g moves all the points of B. The homogeneous spaces $G/\triangleleft A$, where $\triangleleft A$ is the stabilizer of $A$, were called complexions by Planck and Boltzmann. Their "dimensions" (more or less) correspond to the "entropies" studied in information theory. (The latter subject, which soon grew into one of the most successful mathematical theories of all time, was only introduced by Shannon in 1948, but it could have been introduced much earlier, all the tools being available by 1870 at the latest!)

The "easy half" of Galois's theorem says that the problem of expressing the roots in terms of radicals is solvable only if the Galois group is solvable (has a composition series, a chain of subgroups each normal in its parent supergroup). With some changes in language, this is true in a much more general context than the theory of fields. The "nontrivial half", the part essentially exploiting the theory of field extensions, is the converse: if the Galois group has a composition series, one can write the roots in terms of nested radicals, with the nesting and the degrees of the radicals corresponding to the factor groups of the composition series, certain prime cyclic groups. (I am oversimplifying a bit.)

Even today, few books on Galois theory attempt to explain how this works--- I forgot to say in my previous post that while the book by Gaal doesn't discuss how to compute the Galois groups of specific polynomials, it does discuss the nontrivial half of Galois's paper, which outlined a construction. (This paper is translated in Edwards, Galois Theory, and as you will see it is quite sketchy--- nonetheless, everyone agrees that all the basic ideas which are required are presented there in embryonic form.) I'll just add that to prove the hard half, Galois brilliantly exploited one of Gauss's greatest innovations, modular arithmetic. Between them, Gauss and Galois are responsible for the fact that so many algebra books contain frequent repetitions of the phrase "reduce modulo a prime".

Last but not least, in his very first paper, Galois (IIRC he was 17 at the time) proved something fundamental about continued fractions. See for example the books by Khintchin or Olds. BTW, simple continued fractions turn out to be closely related to several things I hinted at: SL(2,C) is a basic example in theories of representations, invariants, reflection groups, and Lie theory, and the simple continued fraction algorithm is closely related to the euclidean algorithm, which is one of the fundamental
"algebraico-geometrical" algorithsm which is generalized by reduction modulo a Groebner basis. The latter technique (introduced as recently as 1965, although a special case was used in 1900 by Gordan, Hilbert's "rival" in the classical theory of invariants) has reinvigorated algebraic geometry in our times by providing tools to easily and naturally compute with ideals.

"Differential Galois theory" is a somewhat ambiguous phrase, but, yes, the theory developed by mathematicians such as Liouville, Picard, Vessiot, and Lie, which shows that certain elementary indefinite integrals cannot be expressed in terms of elementary functions, was directly inspired by Galois theory. This theory concerns what we could call the Galois theory of a special kind of ordinary differential equation, a Fuchsian equation, in which we exploit the algebraico-geometric structure of certain spaces of functions. Here, the monodromy group plays a role analogous to the Galois group in Galois theory; in fact, as mathwonk mentioned, sometimes one can use one to compute the other.

Liouville, BTW, was the mathematician who was probably the first person other than Galois himself to understand what Galois had done, and fortunately he took the time to explain it to others, by writing the first algebra textbook containing an exposition of Galois theory! This book had a life-altering impact on both Klein and Lie. I should add that Lie is responsible for (mostly rather different) development involving differential equations which was also directly inspired by "the spirit of Galois theory". As for Fuchs, his name is absent from the above list because according to legend, Poincare was once so annoyed by a mathematician by this name who claimed credit for something HP had done, that he named something completely different after Fuchs, just to embarrass poor Fuchs, who was obliged to spend the rest of his life disavowing any knowledge of "Fuchsian equations"!

Even more important was the effect of Galois theory of Klein and Lie, who worked together very closely c. 1870, soon after Galois theory had been rediscovered and explained to other mathematicians by Liouville. Indeed, modern scholars feel that the Erlangen program should perhaps be viewed as a collaboration between Klein and Lie. While Klein devoted much of the rest of his career to working out examples of the correspondence between geometries and "invariants" of group actions, Lie was particularly inspired by the idea of solving (systems of) ordinary and partial differential equations by finding and exploiting their "internal symmetries". He spent the rest of his career working out this great idea, with some neccessary detours to develop what we know call the theory of Lie groups and Lie algebras, which was required background material which didn't exist, so Lie had to invent it. I hardly need explain the important of Lie theory to modern mathematics and physics, but I will say a bit more about Lie's theory of symmetry of differential equations.

This theory falls rather neatly into two parts. In the case of systems of ODEs, Lie showed how we can find internal symmetries and exploit them to reduce the order of the ODEs, say from second order to first order (classically, this process is called "finding a first integral"). In the case of systems of PDEs, Lie showed how we can find internal symmetries (remarkably, while the original system can be highly nonlinear, Lie's equations determining the Lie algebra of symmetries is linear) and exploit them to reduce the number of variables, perhaps even turning a system of nonlinear PDEs into a system of ODEs, which can then be attacked using the first part of his theory.

Lie's theory of symmetries of ODEs underlies most of what symbolic computation packages like Maple do when they solve differential equations symbolically. As Lie showed, most techniques of solving ordinary differential equations which students learn even today in "cookbook courses" are in fact examples of exploiting symmetries of ODEs. And Lie's theory of symmetries of PDEs provides essentially the only known general approach to nonlinear PDEs. It has proven particularly useful in the theory of solitons. I might mention that it has a close connection with Noether's theorem relating symmetries of PDEs to conserved quantities, which allows us, given say a nonlinear "wave equation", to write down notions of "energy", "momentum" and perhaps additional quantities, which will be "conserved" in the sense that the energy of solution, evaluated at some time, remains valid for all later times.

All of these things flowed from just two papers by Galois (or three, depending upon how you count), both shorter and even more cryptic than most arXiv eprints. It's little wonder that the first people to read them, including the great mathematician Cauchy, were utterly baffled by what Galois was trying to say. Fortunately, Liouville was able to figure it out, once he had redisovered some of the ideas first hinted at by Galois in his 1831 paper.

Mathwonk already took a stab at another good question: "what kind of symmetry of the roots of a polynomial is captured by the Galois group?" He sketched very nicely the basic ideas of the discriminant of a polynomial, i.e. the product of the pairwise differences of the roots
$$\Delta = \prod_{1 \leq i < j \leq n} \left( x_i - x_j \right)$$
This is a very important example of a polynomial invariant of a finite matrix group. In embryonic form, we can already see here the essential ideas of the theory of (polynomial) invariants of reductive groups.

Consider the symmetric group $S_n$. One can "linearize" its natural action on a set of n elements by representing each permutation as an n by n matrix whose rows (columns) each contain exactly one 1 with the other entries being 0. As was shown by Newton, the ring of polynomial invariants of this action, i.e. the ring of symmetric polynomials, is generated by the elementary symmetric polynomials:
$$f_1 = x_1 + x_2 + \dots x_n$$
$$f_2 = x_1 \, x_2 + x_1 \, x_3 + \dots x_{n-1} \, x_n$$
through
$$f_n = x_1 \, x_2 \dots x_n$$

More generally, the invariant ring for any subgroup of the symmetric group (has a direct decomposition as the sum of subspaces of homogeneous invariants of degree d, for each nonnegative integer d. The Hilbert series counting the dimension of these vector subspaces of the ring of polynomial invariants of a matrix group can be computed by a handy formula due to Molien, which involves simply averaging certain expressions over the permutation matrices in the group. (This is related to averaging sizes of fixsets--- if you know about the Cauchy-Burnside formula in elementary group theory, this should sound familiar.) For S_3 we obtain from Molien's formula
$$\frac{1}{(1-z) \, (1-z^2) \, (1-z^3)} = 1 + z + 2 \, z^2 + 3 \, z^3 + \dots$$
and for A_3 we obtain
$$\frac{1+z^3}{(1-z) \, (1-z^2) \, (1-z^3)} = 1 + z + 2\, z^2 + 4 \, z^3 + \dots$$
So in the case of A_3, we have an "extra" homogeneous polynomial invariant in the vector basis for the subspace of homogeneous invariants of degree three. We can take this to be the discriminant
$$\Delta(x_1,x_2,x_3) = \left( x_1-x_2 \right) \, \left( x_2 - x_3 \right) \, \left( x_1 - x_3 \right)$$
Moreover, writing the invariant ring of S_3 as R and that of A_3 as S, we have
$$S = R \oplus \Delta \, R$$
which adumbrates the all-important Cohen-Macaulay property.

Similarly, the other subgroups of the symmetric group have invariant rings for which in the Molien series, the numerator lists (additively) the number of secondary invariants by degree, while the denominator lists (multiplicatively) the number of primary invariants by degree. The primary invariants are the generators of R, namely the elementary symmetric polynomials. A fundamental fact proven by Hilbert is that the there exist no algebraic relations among the primary invariants (no polynomial function of them vanishes identically), but the secondary and primary invariants do admit such relations, which were called syzygies by Sylvester. A fundamental task of commutative algebra is to compute the syzygies (and the syzygies of the syzygies--- another of Hilbert's great theorems states that this process terminates after finitely many steps). Noether's transfer operator is a R-module homomorphism from the R-module S to R, and in each dimension its kernel gives the polynomials which are invariant under the subgroup but not the full symmetric subgroup. The Cohen-Macaulay property is essentially the fact that S is a finitely generated free R-module.

With minor changes, these facts are true for other reductive groups, including reflection groups.

In the case of Galois groups, in the generic case the only invariants are the primaries. We don't know the roots but we do know that the values assumed by the primaries when we plug in the roots for $(x_1, x_2, \dots x_n)$ are simply the coefficients of the polynomial, which are rational. Sometimes the Galois group is smaller, a proper subgroup of the symmetric group. That happens when the polynomial factors (in which case the Galois group doesn't act transitively). More interesting is the possibility that the Galois group is a transitive subgroup of the symmetric group, which happens if the polynomial is irreducible but has some "extra invariants" corresponding to "unexpected asymmetries" in (algebraic combinations of) the roots. In this case, the extra invariants give certain polynomial combinations of the (unknown) roots which happen to be rational.

This is particularly easy to understand in the case of the discriminant, where the syzygy is a quadratic expression in the discriminant (secondary invariant) and the primary invariants. To wit, the square of the discriminant is a certain polynomial function of the primary invariants (or if you prefer, of the coefficients of the polynomial we are investigating). More generally, one can write down syzygies for the invariant rings of the other transitive subgroups. Plugging in the value of the unknown roots, we obtain polynomial expressions (incorporating the coefficients of the polynomial) which may be reducible; if so, we have found an extra invariant. Combining such information about the ring of invariants, one can try to determine the Galois group directly. This is the essential idea behind Newton's method of computing Galois groups: find a collection of witnesses, interrogate them, and figure out who must be reponsible for the pattern of evidence so obtained. (In the modern literature, this idea of Newton has developed into a circle of ideas using Schubert polynomials, a generalization of Schur polynomials, which give a basis of the vector subspace of homogeneous symmetric polynomials of degree d, as in the Hilbert decomposition of the invariant ring of the symmetric group.)

Sturmfels disavows Newton's approach in his book, without explanation, but I guess the reason may be that computing syzygies quickly becomes notoriously demanding on computational resources; other approaches to computing the Galois group are far more efficient. Still, Newton's way of understanding Galois groups is perhaps the most direct route to understanding, and it fits them into a rather grand picture.

Returning to the Kleinian theme, one can now attempt to interpret the algebraic relations implicit in our polynomial invariants in geometric terms. In his book, Sturmfels discusses this in some detail for complex projective spaces and for euclidean spaces, but there is much, much more to say.

Last edited: Aug 19, 2007
19. Aug 18, 2007

bham10246

Is this an excerpt from a book? Wow...

So basic algebra is very important as we continue to study mathematics.

By the way, your entry is pretty interesting...

20. Aug 19, 2007

bham10246

Hi Chris Hillman,

I didn't mean it in a bad sort of way. I just thought you are one of the many people on this forum who leaves behind great stuff to think about, gives wise advices to students, and am able to combine different fields to help us to keep a bigger picture in mind.

For example, often I ask myself: why am I studying algebra, why do I need Galois theory, how does it help our country and other countries, how does all this pure mathematics (algebra, analysis, functional analysis, ODE, PDE, geometry, algebraic geometry) help other people?

This is something I would like to discover within the next few weeks because I need to sign up for fall classes. However, there are so many classes to choose from: Hilbert spaces, harmonic analysis, Complex Analysis II (Dirichlet problem, picard theorem, quasiconformal mappings, Beltrami equations, Beurling-Ahlfors extensions), algebra, Lie groups, algebraic curves and surfaces, probability, combinatorial mathematics, extremal graph theory, differential geometry, logic, descriptive set theory.... I am still not certain which classes to take!

I'm at the point of asking myself: in which areas is mathematics research the most active? Which classes should I take so that I have, not only teaching/professorship opportunities, but I also have opportunities to work with the government and in industry (like designing cars, aerodynamics, oceanography, waves, electronics, designing faster and more efficient computers, coding for computer security, etc)?

I met some people over the summer and they asked me what's pure mathematics? Nice and clean mathematics? What do you study?

Although I am a student in mathematics, the most I could tell them is restate some of the theorems because even I don't have a general overview of how pure mathematics is beneficial to them and what lies behind each course.

So in summary, I'm glad there are people like you who has so much to share with people like me, because I too can go forward and tell others great things about mathematics. And of course, I will tell them that it's from people at Physics Forum.

Last edited: Aug 19, 2007