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Results tagged with finite-fields
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user 2530
A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.
14
votes
1
answer
783
views
Noether-Deuring for injections and surjections?
Noether-Deuring theorem (not in the strongest form, but in the one I usually need):
Let $L\diagup K$ be a field extension. Let $A$ be a $K$-algebra which is finite-dimensional as a vector space over …
- 33.8k
13
votes
Cubic polynomials over finite fields whose roots are quadratic residues or non-residues
Yes, it does. Here is a stronger claim:
Theorem 1. Let $F$ be a field of characteristic $\neq2$. Let $c\in F$. Let
$r\in F$ be a nonzero square, and let $n_{1},n_{2}\in F$ be such that the
pol …
- 33.8k
11
votes
3
answers
498
views
Local-globalism for similar matrices?
My background on number theory is very weak, so please bear with me...
Given two matrices $A$ and $B$ in $\mathbb{Z}^{n\times n}$. Assume that for every prime $p$, the images of $A$ and $B$ in $\math …
- 33.8k
8
votes
Accepted
Noether-Deuring for injections and surjections?
Hendrik W. Lenstra has just informed me that the answer to my question is "no", both in the case of $r = \dim U$ (so we are looking at injective $A$-linear maps) and in the case of $r = \dim V$ (so we …
5
votes
1
answer
598
views
Are there Carlitz analogues of quadratic residues and reciprocity?
Let $q$ be a prime power. I will use the notations of Keith Conrad's Carlitz extensions paper (but I'll work over $\mathbb{F}_q$ rather than $\mathbb{F}_p$).
The most general question I'm asking here …
- 33.8k
3
votes
Accepted
Proving inequation with ceilings in Finite Field of characteristic $p$
We have $1\equiv u\left( p-r\right) \equiv u\left( -r\right)
=-ur\operatorname{mod}p$, so that $p\mid1+ur=ur+1$. Hence, $\left\lceil
\dfrac{ur}{p}\right\rceil =\dfrac{ur+1}{p}$.
We need to prove t …
- 33.8k
3
votes
Accepted
Conjugation in associative algebras over finite fields
$\newcommand{\End}{\operatorname{End}}$
$\newcommand{\op}{{\operatorname{op}}}$
The finiteness of $F$ is not needed. I have learnt the idea of the following proof from Torsten Ekedahl.
In the followi …
- 33.8k
3
votes
How to factorize X^n - 1 in Z/pZ?
This seems very much like homework to me, so I'll be brief. I assume that your $Z_p$ denotes the field with $p$ elements; I will call it $\mathbb{F}_p$ henceforth (lest it be confused with the ring $\ …
- 33.8k