Recent content by mathwonk

@Hornbein: they give us the syllabus, and the criterion for passing, and then they tell us we cannot fail the number of students who actually would fail by those criteria, even if those students do not attend either class or office hours, nor turn in homework. It poses a difficult challenge.

Feel free to ignore, as I am interpreting this question more generally as "should we teach in college things that students should have learned earlier?, i.e. should prerequisites matter?" My take: For most of my 40 year career as a college math professor, I proceeded somewhat as follows...

Ok, going down same rabbit hole as Edwards, trying to explain Galois in modern terms, in particular, why the table at the bottom of page 916 of the linked AMS Notices article, which is on page 116-117, of Neumann, does display the Galois group of the equation g whose roots are a,b,c…...

I have spent some time pondering the first part of the translation of Galois' "first memoir" by Neumann, roughly pages 107-117 in Neumann's, The mathematical writings of Evariste Galois, from the European Mathematical Union, and comparing it to H. Edwards AMS Notices exposition. I confess I...

the difference between someone who understands Galois theory and the rest of us is illustrated by this simple question: when asked why the sum of a complex number and its conjugate is real, I say because (a+bi) + (a-bi) = 2a which is real; and why the product is also real: well, (a+bi)((a-bi) =...

wouldn't an AI bot have more perfect grammar?

Please forgive me for reminding you that it is much easier to make a list than it is to read profitably through that list. In my opinion, the books on your list would require many years of hard study for a very bright focused person to actually master. Moreover, I doubt if anyone I know has...

It is true that OP has not checked simple connectedness. However, although the theorem stated specifies simply connected, all that is really needed is for the set D not to contain a path that winds around the point p, which I take to be the meaning of the OP's requirement that the set not...

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...