All Questions
Tagged with finite-fields fields
23 questions
2
votes
1
answer
232
views
Chinese remainder theorem for composition
Let $f(x) \in F_p[x]$ and I know $f(x)$ modulo two polynomials $\phi_1(x)$ and $\phi_2(x)$. What sort of information about $f$ modulo the composition $\phi_1(\phi_2(x))$ can I recover?
I'm looking ...
- 591
1
vote
1
answer
285
views
A "semi-genetic" definition of addition and multiplication in the field $\operatorname{On}_p$?
Let $+,\cdot$ denote multiplication in $\mathbb{N}_0$. The addition and multiplication in $\operatorname{On}_p$ are denoted $\oplus, \otimes$.
Recursive definition of addition:
$$x \oplus y := ((x+y) \...
- 3,064
10
votes
1
answer
327
views
Proving that polynomials belonging to a certain family are reducible
In an article, I've found the following result. Unfortunately, it was derived from a general, somewhat complicated theory, that would be cumbersome for this result alone.
Assume that $\mathbb F_p$ is ...
- 687
1
vote
2
answers
391
views
A quantity associated to a field extension
Let $F\subset E$ be a field extension. So $E$ has a natural structure of $F$-vector space.
A vector subspace $V\subset E$ is a special subspace if $F\subset V$ and $V$ is closed under the inverse ...
- 356
2
votes
0
answers
92
views
System of linear equations in positive characteristic
Let $K$ be a field of positive characteristic $p$. Consider the system of $\mathbb F_p$-linear equations
$$\left\{\begin{array}{ccl}
a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n&=&b_1\\
a_{11}x^p_1+a_{...
- 3,998
0
votes
0
answers
79
views
Does "tensoring" with a fixed field preserve Galois extensions of finite fields?
Let $K$ be a (possibly infinite) field of characteristic $p$, and $L$ be a finite field extension of $\mathbb{F}_p$, so that $L$ is finite and $L/\mathbb{F}_p$ is Galois. Suppose $K \otimes_{\mathbb{F}...
- 1,328
2
votes
0
answers
67
views
Partitioning a given set into root set of two polynomials or non zero set of two polynomials simultaneously
Let $p,q$ be two prime numbers and let $p<q.$ Also let, $\mathbb{Z}_p$ and $\mathbb{Z}_q$ denote the fields formed by integers modulo $p$ and modulo $q$ respectively (with respect to the modulo $p$ ...
- 850
2
votes
0
answers
243
views
Sums of squares in fields
Which fields $k$ have the property that any sum of squares is a square ?
Are there elegant characterizations and/or classifications known for this type of field ?
And what if we replace "fields" by "...
- 4,547
1
vote
0
answers
140
views
Nature of polynomials of the form $x^n-a$ over finite fields
I state the following theorem from Serge Lang's Book- Algebra(3rd edition).
Theorem: Let $k$ be a field and $n$ an integer $\geq$ 2. Let $a\in k, a\neq 0$. Assume that for all primes $p$ such that $...
- 428
1
vote
1
answer
2k
views
Write the algebra closure of $F_p$ as union of finite fields [closed]
This question follows Field theory by Steven Roman, Chapter 9, Exercise 20.
Denote the algebraic closure of the finite field $F_q$ by $\Gamma(q)$, and let $a_n$ be any strictly increasing infinite ...
- 205
4
votes
6
answers
642
views
Alternate descriptions of finite fields
The finite field of order $p^n$ is isomorphic to $(\mathbb Z/p \mathbb Z)[X]/(P)$, where $P$ is an irreducible polynomial in $(\mathbb Z/p \mathbb Z)[X]$ of degree $n$. This describes every finite ...
- 6,353
13
votes
1
answer
603
views
Steinberg representation for sporadic simple groups?
The Steinberg representation is a remarkable irreducible representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-...
- 24.9k
5
votes
1
answer
204
views
Is it true that every subspace contain a primitive element?
Let $R = GF(q), q = p^r$, be a field with identity $e$, where $p$ is a prime number. Let $S=GF(q^n)$ be an extension of $R, n\geq 2$ and $K = GF(q^{mn})$ be an extension of $S$, where $m$ is prime. ...
4
votes
0
answers
247
views
Polynomial equations in many variables have solutions (Lang 1952 paper)
I am trying to understand the proof of the following result:
Suppose $F$ is a function field in $k$ variables over an algebraically closed field. Let $f_1,...,f_r \in F[x_1,...,x_n]$ be ...
- 87
6
votes
2
answers
462
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\...
3
votes
0
answers
290
views
Fields whose algebraic closure is a finite extension [duplicate]
It is well-known that the complex numbers $\mathbb{C}$ is a degree two extension of $\mathbb{R}$, where one possible minimal polynomial is $x^2 + 1$. Further, $\mathbb{C}$ is algebraically closed.
...
- 26.9k
2
votes
1
answer
333
views
Uncountable cardinals and Prufer $p$-groups
Let $A$ be an elementary Abelian uncountable $p$-group. Is it known if there is an action of a Prufer $q$-group (here $q$ is a prime not necessarily distinct from $p$) $C_{q^{\infty}}$ onto $A$ such ...
- 599
1
vote
0
answers
228
views
How can I decode efficiently a triple-error-correcting binary BCH code?
In a given $\mathrm{BCH}(N,K)$, $T=3$ code over $\mathrm{GF}(2^m)$, there are ways to find the error locations in a given $N$-bit codeword directly from the syndromes without going through the normal ...
- 87
1
vote
1
answer
3k
views
Generators of cyclic group of finite fields
Let $F$ be a finite field and $f(x)\in F[X]$ be an irreducible polynomial of degree $n$. Let $\alpha$ be a root of $f(x)$. So $E:=F[\alpha]$ is a finite field of order $|F|^{n}$.
We know that $(E^{\...
- 461
56
votes
14
answers
21k
views
Fantastic properties of Z/2Z
Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...
9
votes
4
answers
1k
views
The "interplay" between additive and multiplicative structure in a field
A field is an ordered triple $(F, +,\cdot)$ of a set $F$ and binary operations $+,\times$ on $F$ such that $(F,+)$ and $(F\backslash 0,\times)$ are abelian groups satisfying the distributive laws
$\...
- 2,075
11
votes
3
answers
1k
views
Can we always find such an irreducible polynomial of degree n where degree(p(x)-x^n)<= n/2?
Consider unitary polynomials of degree $n$ over $GF(2)$. That is, polynomials of the form $p(x) = \sum_{i=0}^n a_i x^i$ where $a_i\in GF(2)$ and $a_n=1$.
Can we always find such an irreducible ...
- 375
4
votes
3
answers
835
views
Field with cyclic product group
If a field has a cyclic multiplicative group, is it necessarily finite?
- 1,441