Recent content by mathwonk

The natural logarithm is by definition any analytic function defined on a (connected) open set U not including 0, and (right-) inverse to the exponential function. (I.e. such that exp(Ln(z)) = z for all z in U.) Such inverses do not exist on open sets which include a loop that goes around the...

I currently still have over 250 mathematics, and some more physics, books from my university career on my (sturdy built-in) shelves , having given away roughly an equal number when I retired and moved, from difficulty of transferring them. I regret every one of those lost volumes, and wonder...

Re-reading Mike Artin's book, Algebra, now after many years, I am struck by how very clear his explanations are. It may help that he was writing for college sophomores, although of course MIT sophomores. Anyway, now that I have thought about this stuff again, and read Galois, and gotten in my...

Thanks martinbn, I enjoyed perusing this paper from Keith Conrad; I thought his examples were particularly enlightening. https://kconrad.math.uconn.edu/blurbs/galoistheory/galoisaspermgp.pdf

Briefly (after reading Neumann's translation of Galois): We say a group G can be reduced by a prime p, if G has a normal subgroup N of index p, and we regard G then as reduced by p to the normal subgroup N. In particular the order of N has one fewer prime factor of p than does G. Then a...

A classic example: e^(a+b) = e^a.e^b, and as consequences of this (since e^iz = cos(z) + isin(z)), the sin and cosine formulas. These "addition formulas" interested a classmate of mine at Brandeis in the late 60's, and he wrote a PhD thesis on what functions satisfy such addition formulas. I...

Summary of statements of Galois theory (in characteristic zero): [I.e. assume all fields contain Q, the rational numbers]: Let F be a field, K an extension field of F, and define the automorphism group G(K/F) of K over F, as the group of all automorphisms of K leaving points of F fixed. Assume...

Here is my summary of Galois theory after reading Stewart's brief reproduction of Galois' own words. An approximation of Galois’ own version of his results (from reading Stewart): Let f be a polynomial of degree n with coefficients in a field F, and let K be the field generated over F by the...

the same question was asked 4 years ago, and answered there: my answer is post #7: https://www.physicsforums.com/threads/any-recommendations-for-good-resources-for-learning-galois-theory.1003976/

A few words about symmetry Solving a polynomial equation f with coefficients in F, means writing the roots of f in terms of the coefficients of f, and elements of F. This is inverse to the problem of writing the coefficients in terms of the roots, something easy to do, and also enlightening...

EDIT (added later): After consulting Stewart, Galois theory, 4th edition, I see he includes a translation of Galois' own statement of his main theorems. In particular even his definition ion then "Galois group" is different (in emphasis) from the modern one I gave below, and all other authors...

Those notes 844-1, 844-2, of mine linked in post #12, have as prerecquisite the earlier notes 843-1, 843-2, on the same webpage: https://www.math.uga.edu/directory/people/roy-smith The notes 843-1 are on groups, the notes 843-2 use groups and specifically Galois groups to give a necessary...

apparently (by Smale's h-cobordism theorem) the crux is the pseudo-isotopy class of the (orientation preserving) gluing map: https://math.stackexchange.com/questions/1807683/what-is-the-relationship-between-diffeomorphisms-of-the-sphere-modulo-isotopy-an apparently the equivalence classes of...

some arithmetic: according to AI, there are about 50 million public school students in the US, and to extend the school year by 10 weeks (2 week summer vacation instead of 12 weeks) would cost almost $4,000 each, or almost 200 billion extra dollars. that's assuming the teachers and staff are...

my wife came back from the fancy store- with- everything today with a bag of polenta, and a photo of the bag of "gluten free polenta" next to it, priced at a dollar more. then she reminded me that all polenta is gluten free, being corn. The two bags had the same brand name and same ingredient...