0
votes
1answer
39 views

Multiplying in GF(128)

I know that in GF(128) $a + b = a \oplus b$. I have an multiplication table for GF(128). In this table $7\cdot 5 = 27$. How can I create table like this?
1
vote
1answer
27 views

Is this function part of the Galois group?

Let $\Bbb K$ be a Galois extension of $\Bbb F_p$, where $p$ is prime. Let $\Phi$ be a function on $\Bbb K$. If $\Phi$ is an automorphism of $\Bbb F_p$ that permutes the roots of the minimal ...
5
votes
2answers
63 views

Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the ...
2
votes
1answer
34 views

Galois field of order $p^n$

Can someone help me in establishing an image of how the group looks like. I am having a hard time visualizing it.
0
votes
1answer
39 views

find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
5
votes
0answers
56 views

The reducibility of polynomial $x^{2^n}+x+1$ over $Z_2$ [duplicate]

I meet the problem in Hungerford's Algebra, page 282 as follows: If $n>2$, then the polynomial $x^{2^n}+x+1$ is reducible over $Z_2$ . I have no any idea on this. But I really want to know how to ...
2
votes
1answer
60 views

Relation between divisibility of polynomials in different rings, $h | f$ in $\mathbb{Z}[x], \mathbb{Z}/p^k\mathbb{Z}[x]$ and $\mathbb{F}_p[x]$

Let $p$ be a prime, $k$ a positive integer. Let $f,h \in \mathbb{Z}[x]$ be polynomials such that $h | f \mod p^k$ in $ (\mathbb{Z}/p^k\mathbb{Z})[x]$ $h \mod p$ is irreducible in $\mathbb{F}_p$ ...
1
vote
1answer
46 views

The degree of an irreducible polynomial divides an integer n.

Let $F$ be a finite field such that $|F|=q$, and $f(x)\in{F[x]}$ is irreducible with $deg(f)=d$, $g(x)=x^{{q}^{n}}-x$. Then prove that $f$ divides $g$ if and only if $d$ divides $n$.
-1
votes
1answer
45 views

Galois Field $2^3$ Multiplication of $111$ and $111$

In Galois Field Multiplication is performed as a binary multiplication between the two binary values representing polynomials and then is reduced by a reduction polynomial So if i have the binary ...
1
vote
1answer
26 views

Cyclic Codes over $GF(q)$

Does the set of cyclic codewords / codeword polynomials themselves form a field ? I think they donot because the modulo operation is with respect to $x^n-1$ which is not a prime polynomial. Also the ...
0
votes
0answers
37 views

Prime polynomials over GF(q)

Could anyone please throw some light on my questions? What is the splitting field of an $m$th-degree prime polynomial over $GF(q)$? Is it always $GF(q^m)$? How many $m$th-degree prime polynomials ...
1
vote
2answers
41 views

Proving the roots of a polynomial are a subfield

Define $f(X)=X^{p^r}-X \in \mathbb{F}_p[X]$ where $r\geq 1,\ p$ prime. I'm to show its roots are a subfield of its splitting field $E$ (I have already shown it has $p^r$ distinct roots in $E$). What ...
3
votes
0answers
45 views

Question about why $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$.

I'm trying to follow a solution I'm reading. The idea is to prove $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$ when $n=1$ or $n=p=2$. The solution is as follows: If $x^{p^n}-x+1$ is ...
4
votes
1answer
82 views

My proof: Frobenius Map generates $\mbox{Gal}(\mathbb F_{p^n}/\mathbb F_p)$

I would like to ask, whether anyone can confirm or correct the following version of the proof that the Frobenius map generates the Galois Group of a finite field. Proof First note that ...
4
votes
1answer
97 views

Is this proof that $\mathbb F_q^*$ is cyclic correct?

I would like to show that the multiplicative group of a finite field is cyclic. My goal is to use the best of my knowledge to render the argument as efficient yet simple and understandable as ...
0
votes
1answer
96 views

Finite Fields: check my description/derivation

I am preparing for my exam in Advanced Algebra and Galois Theory, and I am trying to find an efficient way to communicate main properties of Finite Fields. If someone could check my approach and ...
2
votes
3answers
92 views

Compute the Galois group over $F_{101}$

The problem is as follows: Determine the Galois group of the polynomial $f(x)=x^4-2$ over the finite field with $101$ elements, $\mathbb{F}_{101}$. I am not really sure how to go about this, but ...
1
vote
0answers
31 views

Splittinf field of a product of irreducible polynomials over Finite Fields

I was wondering if someone knew a reference, that I could look up, of what I think is a fact that: If $f(x), g(x)$ are two irreducible polynomials in $\mathbb{F}_p[x]$, for $p$ a prime, of respective ...
2
votes
1answer
83 views

Galois group of $X^4+X^3+1$ over $\mathbb{F}_4$

I'm confused. Realizing $\mathbb{F}_4=\mathbb{F}_2[T]/(T^2+T+1)$, the polynomial $X^4+X^3+1$ splits as $(X^2+TX+T)(X^2+(T+1)X+T+1)$. These 2 factors have no root over $\mathbb{F}_4$, so they're ...
2
votes
2answers
193 views

automorphisms of a finite field

Let $F$ be a finite field of characteristic $p$ over $\mathbb{Z}_p$ with $p^r$ elements. Then I have proved that the map $\phi: x\mapsto x^p$ is a field automorphism of $F$. Moreover, $\phi(x)=x$ if ...
3
votes
0answers
92 views

Ultraproduct of the algebraic closure of finite fields

Ok, I have done a little research on the next problem, and there are simple proofs of it using model theory, logic and the Łoś's theorem. But here, the idea is to prove it using only Field Theory, ...
2
votes
1answer
53 views

Find a finite extension of $\mathbb F_2(x,y)$ having infinitely many intermediate subfields.

Let $\mathbb F_2$ be a field of order $2$. Let $F$ be the function field $\mathbb F_2(x,y)$, i.e., $x,y$ are independent variables and $F$ is the field of quotients of the polynomial rings $\mathbb ...
0
votes
0answers
33 views

$K\supset \mathbb{F}_q$, $h=x^q-x+a$, $a\in K$ if $K$ is finite $h$ is reducible

$q=p^n$, $K\supset \mathbb{F}_q$, $h=x^q-x+a\in K[x]$ if $K$ is finite $h$ is reducible. Let $L$ the splitting field of $h$ over $K$. Attempt: I proved that if $\beta$ is a root of $h$ and $h$ ...
4
votes
1answer
70 views

Two questions on finite fields

I'm having some difficult with finite fields. If someone could point out a direction in which to look for these, or link to relevant material online, I would really appreciate it! I'm asked to factor ...
5
votes
2answers
133 views

Galois group of algebraic closure of a finite field

Is there any element of finite order of the Galois group of algebraic closure of a finite field, and if there is how can I construct it ? Thanks.
3
votes
1answer
194 views

Irreducible factors for $x^q-x-a$ in $\mathbb{F}_p$.

If $\mathbb{F}_q$ is a finite field with $q=p^m$ elements where $p$ is prime, if $a\in\mathbb{F}_p^\times$, ¿Which are the irreducible factors of $f_a=x^q-x-a$ ? Attempt: If $\alpha$ is a root for ...
1
vote
0answers
70 views

On the fields of rational fractions over $\mathbb{F}_{p}$

Let $K$ be a field with characteristic $p>0$. Let $K^{p}=\left\{ a^{p}:a\in K\right\}$. Prove that $K^p$ is a subfield of $K$. Furthermore, if $K=\mathbb{F}_{p}\left(X\right)$ is the fields of ...
12
votes
0answers
108 views

Ring structure on the Galois group of a finite field

Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure ...
0
votes
1answer
77 views

Galois extension

http://www.math.uiuc.edu/~r-ash/Algebra/Chapter6.pdf In the pdf, corollary $6.4.2$, the extension $E/F_p$ is Galois. Why is it Galois? Is it because $F_p$ is a finite field and hence every ...
3
votes
2answers
139 views

What is meant by 'the completion of Z'?

In the first chapter of Algebraic Number Theory (lecture notes collected by Cassels-Fröhlich), page 28 has the following paragraph: "We suppose now that $k$ is a finite field of characteristic $p$ ...
1
vote
0answers
97 views

Determine generator over $GF(2^4)$

Working in $GF(2^4$) Field generated modulus $x^4+x^3+x^2+x+1$. Find a generator of $F$. What I have figured out so far - $16$ polynomials to consider. If $b$ is generator then start with $b = x + ...
2
votes
2answers
88 views

Finding all finite field embeddings

Does there exist an efficient (i.e. sub-exponential) algorithm for finding all possible embeddings of one finite field into an isomorphic field? I'm particularly interested in ways to embed ...
0
votes
0answers
37 views

Dummit exercise 14.3.11: extension degree of finite fields [duplicate]

Dummit and Foote's exercise 14.3.11 asks to prove that $f(x) = x^{p^{n}}-x+1$ is irreducible over $\mathbb{F}_{p}$ iff $n=1$ or $n=p=2$. To prove the 'only if' part, the exercise suggest to prove that ...
6
votes
2answers
138 views

Is it true that every element of $\mathbb{F}_p$ has an $n$-th root in $\mathbb{F}_{p^n}$?

It is not hard to prove that every element of $\mathbb{F}_p$ has a square root in $\mathbb{F}_{p^2}$: take any $a \in \mathbb{F}_p$ and consider the polynomial $f = X^2 - a$. If $f$ has a root in ...
2
votes
1answer
109 views

Irreducible factors over $\mathbb{F}_{2^n}$

The question is this: What is the number and degrees of the irreducible factors of $f(x)=x^{11}+1$ in $\mathbb{F}_{2^n}[x]$, for $n\in \mathbb{N}$? Thoughts: I know that the elements of ...
0
votes
0answers
58 views

$x^p-x-a$ is irreducible over $F_p$ for p-prime, $a\in F_p, a\neq 0$ [duplicate]

Actual Question is to prove that, for a prime p and $a\in F_p$, $a\neq 0$, $f(x)=x^p-x+a$ is irreducible. This is an exercise Question in Dummit Foote 13.5.5. Hint : Prove that if $\alpha$ is a root ...
3
votes
1answer
663 views

Construct a finite field of 16 elements and find a generator for its multiplicative group.

Construct a finite field of 16 elements and find a generator for its multiplicative group. Find all generators of multiplicative group. Very obvious Construction of a field with 16 elements according ...
1
vote
1answer
112 views

Splitting field of $x^m - 1$ over $\mathbb F_p$

I need to find the splitting field of a polynomial $x^m-1 \in\mathbb{F}_p[x]$. I know that if $(m,p)=1$ then the splitting field is $\mathbb{F}_p(z)$ where $z$ is primitive root of unity of order $m$. ...
2
votes
2answers
194 views

Proof of Galois' theorem that there exists a field of $p^n$ elements.

From Galois Theory (Rotman): For every prime p and every positive integer n, there exists a field having exactly $p^n$ elements. Proof. If there were a field K with $|K| = p^n = q$, then ...
0
votes
1answer
47 views

How to convert a polynomial in $GF(p^n)$ into the form $a(x)^k$, with $a(x)$ a generator polynomial.

I basically have two questions: Given $GF(p^n$) and $g(x)$ an irreducible polynomial: My textbook says that a polynomial is called primitive if $x$ is a generator of the field. The question now is, ...
1
vote
1answer
83 views

Finite fields as splitting fields

hey guys so i stumbled upon an example that begins with "Consider GF(25). This can be constructed as the splitting field of $t^2 - 2...$" But the theorem states that it is the splitting field of the ...
0
votes
2answers
113 views

finitely generated subfield of algebraic closure of the finite field with $p$ elements

Let $\mathbb{F}^{\operatorname{alg}}_p$ be the algebraic closure of the finite field with $p$ elements. I know that any finitely generated subfield of $\mathbb{F}^{\operatorname{alg}}_p $ is ...
2
votes
0answers
167 views

Irreducible polynomial roots and representations for Galois field elements in normal basis

I am trying to understand some topics in the literature and came across the following problem. Say I have a field $GF(2^4)$ defined by the irreducible polynomial $r(z) = z^4 + z + 1$, and I want to ...
3
votes
1answer
119 views

Irreducible polynomial over $\mathbb Z/7\mathbb Z$

Is the polynomial $t^6-3$ irreducible over $\mathbb{Z}/7\mathbb{Z}$? How would you prove that without doing all the computations? Is there a general method for this? Thank you.
3
votes
1answer
193 views

Lagrange interpolation of Galois field functions

I'm trying to understand how to apply Lagrangian interpolation for a function $F : GF(2^k) \to GF(2^k)$, but am having some difficulty. Suppose I have a function $F : GF(2^2) \to GF(2^2)$ given by the ...
1
vote
1answer
43 views

Number of solutions of $P(x_1, …, x_n) = 0$ in $\mathbb{F}_q^n$

I have this exercise : $q = p^r$ (it is not clearly precised in the exercise that $r = n$). and $\mathbb{F}_q$ is the finite field of cardinal $q$. Let's $P \in \mathbb{F}_q[X_1, ..., X_n]$ of degree ...
1
vote
0answers
129 views

How does trigonometry in a Galois field work?

This is a follow-up to this question. I'm interested in doing trigonometry in finite fields on a computer. I do not understand precisely how trigonometric functions are supposed to work in a finite ...
10
votes
3answers
403 views

Why does rational trigonometry not work over a field of Characteristic 2?

I am interested in solving triangles in a finite field with a computer program. Rational trigonometry seems well suited to do this. However, the Wikipedia article, as well as several published ...
5
votes
1answer
117 views

Modules for twisted polynomial ring and Galois descent

Let $\mathbb{F}_q$ be a finite field with algebraic closure $\overline{\mathbb{F}}_q$ and consider the twisted polynomial ring $\overline{\mathbb{F}}_q\{ \tau \}$, where multiplication satisfies the ...
6
votes
1answer
322 views

Splitting field of $ x^2 + 1$ over $\mathbb{Z_3}$

I have the following exercise: Find splitting field for the polynomial $x^2 + 1$ over $\mathbb{Z_3}$. My solution: At first, we should try to solve the equation $x^2 + 1 = 0$, thus $x^2 = 2$ ...