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38 votes
1 answer
10k views

Infinite tensor products

Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...
Martin Brandenburg's user avatar
10 votes
4 answers
2k views

Formal power series is Taylor expansion of rational function iff Hankel determinants vanish?

Let $$ u(T)=\sum_{n = 0}^\infty a_nT^n$$ be a formal power series over a field $K$. Then why does $u(T)$ lie in $K(T)$ (i.e. is the Taylor expansion of a rational function) if and only if there is an $...
user avatar
28 votes
5 answers
4k views

Does Smith normal form imply PID?

Let $R$ be a nonzero commutative ring with $1$, such that all finite matrices over $R$ have a Smith normal form. Does it follow that $R$ is a principal ideal domain? If this fails, suppose we ...
user avatar
53 votes
9 answers
13k views

Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$. In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$, in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)...
agt's user avatar
  • 4,306
36 votes
3 answers
2k views

Are large powers of polynomials linearly independent?

Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional. Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?
Guillaume Aubrun's user avatar
34 votes
8 answers
4k views

Uncountable counterexamples in algebra

In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
30 votes
2 answers
2k views

When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?

Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by ...
Ira L's user avatar
  • 418
5 votes
2 answers
2k views

Iterated calculation of determinants

Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
HenrikRüping's user avatar
4 votes
1 answer
728 views

matrix congruence and smith normal form

Fixed $n \geq 2$ and consider $A,B \in GL(n,\mathbb{Z}).$ We know that we have the Smith normal form. One can find $U, V \in SL(n,\mathbb{Z})$ such that $A=UDV.$ So as $B$. The Smith normal form is ...
en kuo's user avatar
  • 145
38 votes
2 answers
6k views

Over which fields are symmetric matrices diagonalizable ?

The question is motivated by this one real symmetric matrix has real eigenvalues - elementary proof: Are there other fields $F$ than $\mathbb{R}$ (maybe some valued fields or real closed fields) ...
tomasz 's user avatar
  • 567
14 votes
0 answers
602 views

Is the Zariski density proof of Cayley-Hamilton circular?

This old MO thread and its comments contains a discussion of the Zariski density proof of Cayley-Hamilton (I have also asked a separate question about the proof Victor gives in the comments here). ...
Qiaochu Yuan's user avatar
7 votes
2 answers
311 views

The coefficient of a specific monomial in the expansion of the following polynomial

Let $a_{n,k}$ be the coefficient of $$X_1^{\frac{k(n-1)}{2}}X_2^{\frac{k(n-1)}{2}}\cdots X_n^{\frac{k(n-1)}{2}}$$ in the expansion of the real polynomial $$\left(\prod\limits_{1\leq i<j\leq n}(X_j-...
user173856's user avatar
  • 1,997
7 votes
3 answers
2k views

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

Is there a field which is the union of finitely many proper subfields?
heiko's user avatar
  • 79
6 votes
1 answer
725 views

Who defined and who coined "module"?

The title of my Q. says it all: QUESTION:   Who defined and who coined: module? Would it be Emmy Noether? EDIT   In view of @anon's and KConrad's answers, and as it could have been ...
Włodzimierz Holsztyński's user avatar
5 votes
1 answer
265 views

Is every matrix involution over a UFD diagonalisable?

Let $A$ be a UFD, that is also a $k$-algebra, where $k$ is a field of characteristic $\not=2$ (for instance polynomials over $k$). Is every involution in $\mathrm{GL}_n(A)$ diagonalisable? This is of ...
Jérémy Blanc's user avatar
4 votes
2 answers
818 views

Double orthogonal complement of a finite module

Crossposted from math.stackexchange since I'm not getting any answer. Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise where $\mathbb{Z}_q$ is the ...
Carl's user avatar
  • 141
2 votes
1 answer
222 views

$C=A \cdot B$ matrices and exact sequence of $DVR$-modules

I'm looking for a proof or a reference for the following statement: Let $R$ be a DVR (Discrete Valuation Ring) and $p$ a prime element, and let $\mathfrak a$, $\mathfrak b$ and $\mathfrak c$ be ...
Maffred's user avatar
  • 291
1 vote
1 answer
367 views

Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial

The following question is motivated by the study of a stability border for a robust linear time-invariant control system. Let us we have an affine family of $n\times n$ matrices with indeterminate ($\...
probably's user avatar
  • 413
0 votes
1 answer
454 views

Conjugacy in the quaternion group

Let $G$ be a non-commutative group, and suppose we are given two elements $x, y \in G$ which are conjugate, i.e. we know there exists some $z \in G$ such that $zxz^{-1} = y$. Can we find $z$ given $x$ ...
Gautam's user avatar
  • 1,703