Is Galois' groundbreaking paper on solvable groups still relevant today?

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In summary, Galois was able to solve a problem that had remained unsolved for centuries by defining a group for polynomials with rational coefficients and showing that the group of a generic n-degree irreducible polynomial is the symmetric group on n letters, S_n.
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jostpuur
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When a Lie algebra is solvable, does it have something to do with actually solving something?
 
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A group, G, (whether "Lie" or not) is said to be solvable if and only if there exist a sequence of subgroups, G1, G2, ..., Gn of G, such that:
G1 is the subgroup consisting only of the group identity,
Gn is G itself,
Each Gi is a normal subgroup of Gi+1, and
Gi+1/Gi is commutative.

The name, and indeed the whole definition, is from Galois's answer to the question of whether or not there could exist a "formula" for polynomials of degree 5 or higher, in terms of radicals.

The point is that a polynomial equation, with integer coefficients, is "solvable by radicals" if and only if its Galois group is a "solvable" group.
I think (I'm entering unsure ground here) that Galois also showed that, for n> 4, there exist a polynomial of degree n whose Galois group is all of Sn (the group of permutations on n objects) and that Sn, for n> 4, is not a "solvable" group.
 
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Quick Clarification

In his 1831 paper, Galois (age 20) defined a group for any polynomial with rational coefficients, a permutation group acting on the roots, which we now call the galois group of the polynomial. He showed (in essence) that this group acts transitively iff the polynomial is irreducible, and if so, it is solvable iff the roots of the polynomial can be expressed in closed form in terms of arithmetic operations plus the extraction of roots (i.e. the problem of finding the roots can be reduced to finding the roots of monomials). In such cases, the roots are expressed as nested radicals, in a manner which mirrors the descending subnormal chain of subgroups (the composition series exhibiting the fact that the group is indeed solvable--- the composition series can be refined so that the quotients mentioned by Halls are in fact prime order cyclic groups, corresponding to monomials of prime degree). In modern terms, he exhibited a galois duality between the subgroups of the galois group and the intermediate extension fields of the extension field of the rationals defined by adjoining the roots of our polynomial to the field of rational numbers. Galois also showed that while "special" polynomials of high degree can have solvable galois groups, the group of a "generic" n-degree irreducible polynomial is the symmetric group on n letters, [itex]S_n[/itex], which is not solvable for n > 4 (indeed its index two normal subgroup [itex]A_n[/itex] is not solvable for n > 4). Thus, there can be no general formula analogous to the quadratic formula for the quintic or higher degrees.

In order to do all this, since groups, rings, and fields (and group actions, and finite projective planes...) did not yet exist, he had to invent them. Indeed, most would agree that he invented modern algebra. All this in a dozen pages, yet his paper clearly contains all the essential ideas, albeit in sketchy and sometimes delphic form.

This paper is often regarded as one of the single most profound advances in mathematical thought, because Galois was the first to clearly see that algebraic objects more complicated than numbers are worthy of recognition and study. In a sense, he took the first step down the path of categorification. As we recently discussed in some other PF threads, he also introduced one of the great themes of mathematics, the notion of symmetry (and its relation to the notion of information).

His remarkable achievement in completely resolving a problem which had remained unsolved for millenia, in an utterly original and completely unexpected manner, inspired many of the greatest nineteenth century mathematicians, including Sophus Lie (whose dream of doing for differential equations what Galois had done for polynomials led to the development of Lie theory as required background for Lie's theory of the symmetries of differential equations). And Galois continues to inspire mathematicians today (one might mention Grothendieck as a more recent example).

For the simplest example of nested radicals, see Cardano's formula for the roots of a cubic:
http://gowers.wordpress.com/2007/09/15/discovering-a-formula-for-the-cubic/
 
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FAQ: Is Galois' groundbreaking paper on solvable groups still relevant today?

1. Why is it important to determine if a problem is solvable or not?

Determining if a problem is solvable is important because it allows us to understand the limitations and potential of our problem-solving abilities. It also helps us to identify which problems are worth investing time and resources into solving.

2. What makes a problem solvable?

A problem is considered solvable if there exists a known method or algorithm that can be used to reach a solution. This means that the problem can be broken down into smaller, more manageable steps that can be solved using logical reasoning or mathematical calculations.

3. Is every problem ultimately solvable?

No, not every problem is ultimately solvable. Some problems may be too complex or require an infinite amount of time or resources to solve. Additionally, there may be problems that we currently do not have the technology or knowledge to solve.

4. Can a problem be solvable but not computable?

Yes, a problem can be solvable but not computable. This means that a solution exists, but it cannot be calculated or determined using any known algorithm or method. This often occurs with highly complex or abstract problems.

5. How do we know if a problem is solvable?

We can determine if a problem is solvable by analyzing its characteristics and comparing it to known solvable problems. We can also use mathematical proofs or empirical evidence to demonstrate the solvability of a problem. In some cases, trial and error or experimentation may also be used to determine solvability.

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