All Questions
9
questions
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177
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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\...
- 913
1
vote
1
answer
98
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 ...
- 913
1
vote
0
answers
1k
views
Are the integers a vector space or algebra over "some" field or over "some" ring?
Every vector $v$ in a finite-dimensional vector space space $V$ of dimension $n$ over a field $F$ has a unique representation in terms of a basis ${\frak B} \subseteq V$, where a basis for $V$ is a ...
- 171
2
votes
0
answers
61
views
Determining Inconsistency of (first-order) Non-linear System of Equations [closed]
Is there a way I can figure out what values of the coefficients of some system of non-linear equations makes the system inconsistent?
Take the following system of equations as an example. The ...
- 21
3
votes
1
answer
205
views
Characterization of Lagrangian planes in symplectic vector spaces over finite fields [closed]
EDIT: As L Spice pointed out, there is an error in the observation. The question is void therefore
Let $p$ be a prime and $q=p^r$. Let $V$ be a $\mathbb F_q$-vector space of dimension four, with a ...
6
votes
2
answers
461
views
Splitting subspaces and finite fields
Hellow. I'm sure that the following is truth, but I can't prove it.
Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and
$A = \{\theta\...
1
vote
1
answer
768
views
Solving Non-Linear Equations over a Finite Field of a Large Prime Order
I want to know is there is an efficient way to figure out whether or not a ( underdetermined) system of non-linear equations have a solution over a finite field of large prime order. The equations ...
- 11
0
votes
1
answer
605
views
Number of Minimal left ideals in the full matrix ring over a finite commutative local ring
Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?
user39204
2
votes
2
answers
361
views
vectors with entries from a finite ring
I've been working recently with vectors over finite fields, but I was hoping to work in a more general setting and consider vectors over finite commutative rings. The question I had is as follows: if ...
- 21