All Questions
Tagged with galois-theory linear-algebra
28 questions
0
votes
0
answers
183
views
Degree 6 Galois extension over $\mathbb{Q} $
Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
- 923
1
vote
1
answer
100
views
Existence of a symmetric matrix satisfying certain irreducible conditions
Let $K$ be a field such that $ \mathrm{char}(K) \neq 2 $. Let $ p(x) $ be an arbitrary irreducible polynomial over $K$ of degree $n$. Using the rational canonical form, we can always construct an $ n ...
- 923
4
votes
0
answers
134
views
Irreducibility of polynomials associated to binomial coefficients
Let $n \geq 2$.
Let $M_n$ be the $(n+1) \times (n+1)$ matrix with entries $\binom{l}{k}$ for $0 \leq l,k \leq n$ and $U_n=M_n + M_n^T$ and let $f_n(x)$ denote the characteristic polynomial of $U_n$.
...
- 26.5k
2
votes
0
answers
159
views
A dimension problem related to an abelian simple extension of a field
$\DeclareMathOperator\Imm{Im}$Let $K=F(\alpha)$ be an abelian extension of $F$ and let $\sigma$ be a map (could be any map) from $K^\times$ (the multiplicative group of $K$) to itself. Define an $F$-...
- 613
1
vote
0
answers
297
views
Surjectivity of a norm map over $ \mathbb{Q} $
Suppose $ (L/L') $ is an galois extension , where both fields are extension of $Q$ of $\dim n $ and $\dim n^{'}$ respectively.Suppose we consider the norm map $ Nr_{(L/L')} :L \rightarrow L' $. ...
- 923
0
votes
0
answers
112
views
Existence of a subspace of having no isotropic 2-plane
Let $V$ be a vector space of dimension $n$ over the field $\mathbb {Q} $. A subspace $W$ is isotropic for a skew-bilinear form $\alpha$ on $V$ if $\alpha(x,y) = 0$ for all $x,y \in W$.
More ...
- 923
2
votes
1
answer
193
views
Irreducible components of a cyclic extension over $ \mathbb{Q} $
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gal{Gal}$Let $ L $ be a cyclic Galois extension of $ \mathbb{Q} $ of degree $ 6 $. So $ G = \Gal(L/\mathbb{Q}) $ is a cyclic group of order $ 6 $. Then ...
- 923
4
votes
1
answer
186
views
The upper bounds on rank $ 2 $ real matrices
Let $ A_{n}(F) $ be the collection of all skew-symmetric matrices over the field $ F $ ($\operatorname{char} F \neq 2 $). Let M be a subspace of $ A_{n}(F) $ such that all non zero elements have rank ...
- 923
1
vote
1
answer
94
views
Problem concerning about an $n$-subspace of $ A_{n}(F) $
Let $A_{n}(F) $ denote the $n \times n$ skew symmetric matrices over a finite field $F$. Suppose $n$ be even and $N$ be a subspace of $A_{n}(F) $. Now if all the non-zero matrices in $N$ are ...
- 923
4
votes
0
answers
113
views
Index of norm $ 1 $ subgroup in a cyclic extension
Let $L/\mathbb{Q}$ be a cyclic galois extension of degree $ 2n $ and $\sigma $ be a generator of $\operatorname{Gal}(L/\mathbb Q)$. Let $ U $ be the collection of all norm $ 1 $ elements of $L^\times$...
- 923
1
vote
0
answers
136
views
Solution of an equation in cyclotomic extension over $ \mathbb{Q} $ of degree $6$
Let us consider a primitive $7^{\text{th}}$ root of unity $\eta$. Then the minimal polynomial of $ \eta $ over $ \mathbb{Q} $ is $1 + \eta +.....+ \eta^{6}$. So the dimension of the $\mathbb{Q}$-...
- 923
3
votes
1
answer
399
views
Maximal common isotropic subspace for a finite family of skewforms
Let $V$ be a vector space of dimension $n$ over a field $F$. An alternating bilinear form $\alpha\colon V \times V \rightarrow F $ will be called a skewform. A subspace $W$ is isotropic for $\alpha$...
- 923
4
votes
0
answers
259
views
On the characteristic polynomial of the Vandermonde matrix
Let $A_n$ be the $n \times n$-Vandermonde matrix (see for example https://en.wikipedia.org/wiki/Vandermonde_matrix )viewed as a matrix over the fraction field of the polynomial ring over a field $K$ (...
- 26.5k
8
votes
1
answer
690
views
Subspaces of $ A_{n}(\mathbb {Q})$ in which all nonzero matrices are invertible
Let $A_{n}(\mathbb{Q}) $ denote the $n$ times $n$ skew symmetric matrices over the rational number field. Let $N$ be a subspace of $A_{n}(\mathbb{Q}) $.
If all the non-zero matrices in $N$ are ...
- 923
6
votes
0
answers
335
views
Galois groups associated to matrices
When $A\in M_n(\mathbb{Q})$, we consider the pencil $A-xA^T$. Then $p_A(x)=\det(A-xA^T)$ is a self-reciprocal polynomial. $p_A$ can only be irreducible if $n=2p$ is even.
Question: For every $p$, does ...
- 3,741
5
votes
0
answers
155
views
"sparsifying" a binary (over the field F2) matrix
Assume I have a matrix $A \in GF(2)$, i.e., $A_{i,j} \in \{0, 1\}$ and the sum is modulo 2. Is there any known algorithms/methods to sparsify (reduce the number of non-zero entries) $A$ while keeping ...
- 183
3
votes
1
answer
77
views
Dimension of fixed vectors of a semi-linear operator
Let $L$ be a field with a field embedding $\sigma:L \rightarrow L$, and $K=L^{\sigma}$ be the fixed field of $\sigma$. For $A \in M_n(L)$ a matrix, consider the set $X=\{x \in L^n|Ax=\sigma(x) \}$ ...
- 6,622
11
votes
1
answer
731
views
Reference request: Volume 2 of Abhyankar's lectures on algebra?
Abhyankar has a magnificent, if meandering (check them out if you want to see what I mean), set of lectures on algebra.
The description:
This book is a timely survey of much of the algebra developed ...
- 111
10
votes
2
answers
2k
views
When is a bilinear form equivalent to a trace form?
Associated to a finite, separable field extension $L/K$, there is a natural nondegenerate bilinear form, the trace form, defined by $$\langle x,y \rangle := \mathrm{Tr}_{L/K}(xy)$$
Now, given a ...
- 1,142
1
vote
0
answers
131
views
A mixed Vandermonde-Wronskian matrix
I am trying to prove that a matrix of the following form is generically nonsingular:
$A= \begin{bmatrix}
1&1&1&1 \\
f_1 & f_2 & f_3 &f_4 \\
(f_1 -\frac{d}{dt}).f_1& (...
3
votes
1
answer
443
views
Is there an algebraic formula for the eigenvalues of a symmetric $n\times n$ matrix?
Is there a formula in radicals for the eigenvalues, or at least the largest eigenvalue, of an $n\times n$ symmetric matrix, in terms of the entries? For what $n$ is there a formula? There obviously ...
- 2,049
6
votes
2
answers
826
views
Galois descent for dimension of vector spaces
Let $L/K$ be a Galois extension (I am interested in $\overline{\mathbb{Q}}/\mathbb{Q}$ so I do not assume it to be finite).
Let $V\subset L^n$ be a $L$-subvector space, of dimension $d$, such that $g(...
- 7,722
2
votes
2
answers
273
views
Need help with paper written in Russian... Yorgov's paper on self-dual codes with automorphisms of odd order
I came across this paper by V.I.Yorgov named "Binary self-dual codes with automorphisms of odd order". (Actually I was first reading another paper by Borello that cited this paper. It later become ...
- 165
5
votes
2
answers
629
views
Is it always possible to "separate" the eigenvalues of an integer matrix?
Call a square matrix Galois-irreducible if all its eigenvalues are Galois conjugates of each other.
Let $M$ be an integer $n\times n$ matrix which is not Galois-irreducible. Is it always possible ...
- 13.4k
0
votes
0
answers
266
views
Finding the effective maximum number of subspaces in a finite dimensional vector space
Hi mathoverflow community, may be some one may give me a hint on the following problem before I spend much time on brute force search.
For $q$ a prime number and $n=6$, let $\mathbb {F}_{q}^{n}$ be ...
2
votes
1
answer
292
views
nth-powers and degree n polynomials with coefficients in field extensions
Hi,
Suppose that $E/F$ is a Galois extension. If $P(X)\in E[X]$ is a (EDIT: monic) polynomial of
degree $n > 0$, such that $P(X)^n\in F[X]$, does it follow that $P(X)\in F[X]$?
Thanks
- 2,842
8
votes
2
answers
746
views
Field extension containing the eigenvectors of a Hermitian matrix
Let H be a (finite-dimensional) Hermitian matrix with algebraic numbers for its entries, all of which lie in some minimal field extension of the rational numbers; call this field ℚ(H) for short. ...
- 2,649
14
votes
3
answers
1k
views
"Conjugacy rank" of two matrices over field extension
I have posted this elsewhere and got only a partial reply. I don't know whether this qualifies the question for an open-problem tag; if it does, please anyone insert it.
Let $L$ be a field, and $K$ a ...
- 33.8k