1
vote
1answer
27 views

Question about the cycle type of the Frobenius Automorphism

Let $f$ be an irreducible and separable polynomial of degree $n$ over $\mathbb{F}_p$. For a finite field $F$ of characteristic $p$, we define $\phi_p:\alpha\mapsto\alpha^p$. Then we know $\phi_p$ is ...
1
vote
1answer
46 views

Field extension of degree 3 and polynomial roots

Deleted the old question, because tho whole question kind of changed. I am facing following problem: Given extension of finite fields $L/K$ of degree $3$, prove that every polynomial of degree $3$ ...
0
votes
1answer
31 views

Constructing common extension of two finite fields

I am facing following problem, and would be very thankful for any input: I am supposed to prove that two finite fields with same number of elements are isomorphic. I know the usual proof via ...
1
vote
1answer
49 views

Proving degree $n$ have at last $n$ roots in $F_q[X]$

How to prove that in $F_{q}[X]$ of degree $n$, have $n$ roots?
1
vote
3answers
51 views

Automorphisms of splitting field of $x^p-x-a$

Let $p$ be a prime and consider the splitting field of $f(x) = x^{p} - x - a$ over $\mathbb{F}_{p}$. I have worked out that the splitting field is $\mathbb{F}_{p}(\beta)$, where $\beta$ is a root of ...
1
vote
0answers
45 views

Galois Group of $x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$

Question is to find Galois group of $f(x)=x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$ What i have done so far is : I could see that $f(x)$ is Irreducible and ...
2
votes
2answers
48 views

Minimal polynomial reducible modulo every prime $p$

Suppose $K = \mathbb{Q}(\alpha)$ with $\alpha = a + b\sqrt{D_1}+c\sqrt{D_2}+d\sqrt{D_1D_2}$ with $D_1,D_2 \in \mathbb{Z}$. Prove that the minimal polynomial $m_\alpha(x)$ for $\alpha$ over ...
0
votes
1answer
36 views

Does the element exist in the Galois Field?

Let p be a prime positive integer and let a be an element of GF(p). Does there necessarily exist an element b of GF(p) satisfying b^2=a? So, taking a element of GF(p), can we find a b element of ...
4
votes
1answer
44 views

The Galois group of the finite field's algebraic closure is not countable

I'm trying to prove that the group $\operatorname{Gal}(\bar F_p /F_p)$ is not countable. My idea is to show that in the sequence $F_p\leq F_{p^2}\leq F_{p^4} \leq \dots \leq F_{p^{2^n}} \leq\dots ...
1
vote
1answer
29 views

Prove for $a,b \in \mathbb{F}_{p^n}$, if $p(x) = x^3 + ax +b$ is irreducible, then $-4a^3 - 27b^2$ is a square in $\mathbb{F}_{p^n}$.

The problem is as the title states. We know that in this case determinant $D = -4a^3 -27b^2$, and also I know that if $G$ is the Galois group of $x^3 + ax + b$, then $$G \subset A_n \, \iff \sqrt{D} ...
2
votes
1answer
77 views

Having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space over $\mathbb{F}_p$.

As the title states, I'm having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space V over $\mathbb{F}_p$. Clearly it is dimension $n$. How can we work with this vector space? ...
3
votes
1answer
55 views

Galois Group of $x^4+x-1$ over $\mathbb{F}_3$

Consider the finite field $\mathbb{F}_3$ and define the polynomial $f(x)=x^4+x-1$ over $\mathbb{F}_3$. I want to find its Galois Group. I observe that $f$ has no root over $\mathbb{F}_3$, so if it ...
1
vote
2answers
82 views

Irreducible Polynomials over a Finite Field

I am motivated by this question, and want to find a solution to the following problem. Question: How many monic, irreducible polynomials of degree $n$ are there over the finite field ...
1
vote
0answers
61 views

Splitting Field of Irreducible Polynomial over a Finite Field

Suppose $f(x)$ is irreducible over $\mathbb{F}_p[X]$ and let $\alpha$ be a root of $f$ in some extension field. I want to prove the following claim. I have included thoughts below, but I am very ...
1
vote
1answer
105 views

Galois field theory

$\mathrm{GF}( 29^2)$ is created by adjoining the root of the irreducible quadratic $p= x^2 + 7x +15$ to the field $\mathrm{GF}(29)$ . The cubic polynomial $q = Y^3 + (26x + 26)Y^2 + (8x+22)Y +13x+23$ ...
0
votes
1answer
140 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?
2
votes
1answer
36 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
83 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
56 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
43 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
57 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
72 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
66 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
57 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
38 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
44 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
45 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
57 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
140 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
104 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
120 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
103 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
37 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
112 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
551 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 ...
4
votes
0answers
105 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
96 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
39 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
91 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
174 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.
4
votes
1answer
229 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
77 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 ...
11
votes
0answers
146 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
98 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
175 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
142 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
118 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
42 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
162 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
118 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 ...