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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

5 votes
Accepted

0 dimensional Dedekind domain?

Historically Emmy Noether's paper introducing the concept "Dedekind domain" certainly included fields (see e.g. Kleiner's book on the history of abstract algebra, which gives axioms). She was characte …
Charles Matthews's user avatar
4 votes
Accepted

Decomposition of finite algebras over finite fields

The general form of the answer is easy to anticipate: a finite commutative ring is artinian. It will be a product of finite local rings, of characteristic that is a prime number. So your question is r …
Charles Matthews's user avatar
4 votes

Is an elementary symmetric polynomial an irreducible element in the polynomial ring?

Doesn't this follow quite quickly by setting one variable equal to 0? Edit: I was thinking this way. Factors of homogeneous polynomials are homogeneous. Setting the final variable $x_n$ to 0 therefo …
Charles Matthews's user avatar
2 votes

Who named it the Snake Lemma?

If you Google for "diagramme du serpent" it becomes plausible that it was a diagram in Cartan-Eilenberg first of all, before a lemma. Interesting example of how Bourbaki became the standard grad stude …
Charles Matthews's user avatar
2 votes

Generators of a maximal ideal of $k[X_1,\cdots,X_n]$

The dimension is exactly n: you are counting linear polynomials modulo terms of higher order. The geometrical interpretation is that intersecting n - 1 hypersurfaces with a common point must give an …
Charles Matthews's user avatar
2 votes

Is there a field which is the union of finitely many proper subfields?

The reason it fails in the case of finite fields is the primitive element theorem (over the prime subfield, even). This is not quite an argument in the general case because of possible inseparability. …
Charles Matthews's user avatar
1 vote

Is $x^p-x+1$ always irreducible in $\mathbb F_p[x]$?

There is kind of an easy proof given that Berlekamp's algorithm works? In the notation of https://en.wikipedia.org/wiki/Berlekamp%27s_algorithm, the point is that the space of polynomials g congruen …
Charles Matthews's user avatar
1 vote

Subrings of rational functions invariant under change of sign

So at the level of function fields, you have a group of order 2 acting on L, the field of fractions of R, and there is a subfield K fixed by it. The extension L/K will be of degree 2, so quadratic. In …
Charles Matthews's user avatar
1 vote

Elementary Luroth theorem proof?

I think this is something Gauss could have proved, and the point is to come up with his sort of proof. I'm not seeing that as too hard. To show a polynomial of degree at least 1 is transcendental over …
Charles Matthews's user avatar
1 vote

cyclic polynomials and their solutions

So you have a cyclic group acting on complex affine space, and the situation at the level of function fields is clear enough: by basic Galois theory the extension of rational functions is a cyclic ext …
Charles Matthews's user avatar
0 votes

Is there much difference between Kronecker's and Dedekind's methods in algebraic number theo...

Urgh - Hilbert stole Kronecker's key ideas on "module theory" to use in invariant theory, while excluding them from his Zahlbericht, and trying to wipe out the "constructive" point of view? Dedekind w …
Charles Matthews's user avatar
0 votes
Accepted

How to design or create or generate a bijective ring map?

In generality (this is tagged "commutative algebra", so let's talk commutative rings) I wonder if there is more than taking generators of each side and writing the images as polynomials in the generat …
Charles Matthews's user avatar