Finite fields are structures arising in abstract algebra. The order of a finite field is always a prime power, and for each prime power $q$ there is a single isomorphism type. It is usually denoted by $\mathbb{F}_q$ or $\operatorname{GF}(q)$.

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I want to show how many intermediate fields there are between $GF(3^{12})$ and $GF(3^4)$.

So $E=GF(3^{12})$ is an extension of $F=GF(3^4)$, and I wish to know how many intermediate fields $F \subsetneq K \subsetneq E$ there are. By a result in Escofier's Galois Theory I have that $G={\rm ...
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3answers
41 views

Finding a basis for a field

I have a polynomial f(x) = $x^3+x^2+1$ in $\mathbb{Z}_5[x]$ and it is given that F = $\mathbb{Z}_5[x]$/$<f(x)>$ = $\mathbb{Z}_5(\alpha)$ where $\alpha =x+<f(x)>$. I want to find a basis ...
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30 views

Irreduciblity of polynomial over field of prime element [on hold]

Let $f(x)=x^4-x^3+14x^2+5x+16$ be a polynomial and let $F_p$ denote the field with $p$ elements, where $p$ is prime number. Then which of the following are always true? Considering $f$ as a ...
2
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1answer
47 views

Galois Theory.Subgroups of Galois Group

How to List the subgroups of the Galois group in general.Im not interested in a specific Example but to make it easier. Supposes the galois group $G=Gal[Q(v,i):Q]$ $$v= \sqrt[4]{2}$$ I know how to ...
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1answer
24 views

What can we say about the dimension of $[ E : F]$ if $F finite$ and $f \in F[x]$ min. polynomial [on hold]

suppose I have a field $F$ and $\alpha \notin F$. $F$ is finite so $char(F)$ is some $p \in \mathbb{N}$ when I have a minimal polynomial $f_\alpha \in F[x]$ with $deg(f_\alpha)=n$ then the dimension ...
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1answer
22 views

Probability that a random polynomial over a finite field can be factorized to linear terms.

Suppose that $f\in\mathbb{F}_p[x]$ is a degree $d$ random univariate polynomial with coefficients from a finite field $\mathbb{F}_p$. What is the probability that $f$ can be written as: ...
3
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0answers
44 views

There is no irreducible polynomial of largest degree in $\mathbf{F}_q[x]$

I am asked to prove or disprove that given a finite field $\mathbf{F}_q$, the ring $\mathbf{F}_q[x]$ contains irreducible polynomials of arbitrarily large degree. I couldn't think of a reason why this ...
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0answers
23 views

Counting the isotropic points for both quadratic and hermitian forms.

Consider an octonion algebra $\mathbb{O} = \mathbb{O}_{\mathbb{F}_{q^2}}$ over a field of order $q^2$, $q = p^k$. Then we have a natural quadratic and hermitean (by this I actually mean hermitean ...
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1answer
27 views

What is the Galois group $Gal(E/F)$ when $F=GF(5^3)$ and $E=GF(5^{24})$

What is the Galois group $Gal(E/F)$ when $F=GF(5^3)$ and $E=GF(5^{24})$. I have attempted to describe the Galois group, but I've become stuck, and it's entirely possible that I've made mistakes as ...
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1answer
23 views

vector space homomorphism for $Map(\mathbb{F}_{5} , \mathbb{F}_{5})$

I'm currently stuck at a mathematical problem and I really don't know where to start.. Since I'm not an expert in Algebra over finite fields... It goes "Define a $\mathbb{F}_{5}$-vector space ...
2
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1answer
57 views

Is the primitive element of an extension of a finite field expressible as a linear combination of adjoined elements?

Suppose $F$ is a field, and $\alpha,\beta$ are algebraic, separable elements over $F$. If $F$ is infinite, the proof of the primitive element theorem gives some $c\in F$ such that ...
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0answers
44 views

Do the odd numbers modulo $2^n$ form a field?

Do the odd numbers modulo $2^n$ form a field (of order $2^{n-1}$) for some $n$? For $n$ a power of 2? If so, this would be quite useful for cryptography.
3
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1answer
52 views

show quadratic forms $x^2 + y^2 + z^2$ and $ x^2 - y^2 - z^2$ are equivalent over finite fields $\mathbb{F}_p$

Can I show the diagonal matrix (1,1,1) and (1,-1,-1) are equivalent over the finite field $\mathbb{F}_3$ Can I show the quadratic forms $x^2 + y^2 + z^2$ and $x^2 - y^2 - z^2$ are equivalent over the ...
6
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1answer
66 views

Matrices and prime numbers

Let $ p $ be a prime number and \begin{align} K=\left\{ \begin{pmatrix} a &b \\ c& d \end{pmatrix} \mid a,b,c,d \in \left\{0,1,\ldots,p-1 \right\}, \right. & a+d \equiv 1 \!\!\!\! ...
2
votes
1answer
29 views

Hilbert function and Hilbert polynomial

I have largely studied Hilbert function and Hilbert polynomial for polynomial rings over fields of characteristic zero. Is it possible to extend the theory also for polynomial rings over fields of ...
2
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1answer
40 views

Is there a simple solution to these 2 equations without trying all possible values?

Now I have two equations and the computation is in a finite field GF(p), where p is a prime. $x, y$ are unknown, and $a, b$ are known. ($0<y<p-1$, and $0<ax<p-1.$) $\begin{cases} x^y ...
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2answers
48 views

Frobenius maps and irreducible functions on finite fields

Let $\mathbb{F}_q$ be a finite field of order $q=p^n$ for some prime $p$ and $n>1$. Suppose both $f(x)=x^2-ax+b$ and $g(x)=x^2-a'x+b'$ are both irreducible. If, assuming that either $a=a'=0$ or ...
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1answer
36 views

Galois field of order 2 constituting a Boolean algebra

We know that the the set $\{0,1\}$ constitutes a Boolean Algebra over the usual $OR$ and $AND$ operations. However, because of the lack of an additive inverse for $1$ this does not produce a Galois ...
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38 views

Is F a field automorphism?

Let $\mathbb{F}_q$ be a finite field of order $q=p^n$ for some prime $p$ and let $f:\mathbb{F}_q\to \mathbb{F}_q$ be a bijection. If $f(0)=0$ (additive identity), $f(1)=1$ (multiplicative ...
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2answers
105 views

Express a prime $p$ as $p=a^2-2b^2$

Suppose $2$ is a quadratic residue modulo $p$ for an odd prime $p$. That is, there is an element $u$ such that $u^2 \equiv 2 \pmod{p}$. From here, can we prove that there exist integers $a$ and $b$ ...
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1answer
66 views

$X^n + X + 1$ reducible in $\mathbb{F}_2$

I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?
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27 views

No $p$-th root implies $X^{p^n}-a$ irreducible for all $n \in \mathbb{N}$

While doing exercises of Chapter IV in Lang's algebra, I encountered the following problem: Suppose char $K=p$. Let $a \in K$. If $a$ has no $p$-th root in $K$, show that $X^{p^n}-a$ is ...
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1answer
29 views

Linear system of equations over $\mathbb{Z}_7$

I had the following set of simultaneous equations in $\mathbb{Z}_7$. $$3x+5y=1$$ $$4x-5y=5$$ Now adding them we get $$7x=6$$ And this has no integer solution in $x$ since $7$ and $6$ are ...
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1answer
36 views

Find all solutions to the equation over field $\mathbb{Z}_{16}$

So for the first part of the question, I have to find the solutions to: $\ x^2+4x+3=0$ over $\mathbb{Z}_{16}$ I have found these to be $X=5,7,13$ and $15$, just by using standard method to solve a ...
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2answers
42 views

Power of a polynomial in Galois field

Let $f(x) \in GF[q](X)$, where $q = p^m$ and $p$ prime. Is the following true? $$f^{p^m}(X) = f(X^{p^m}).$$ I tried to prove the assertion above and got stuck at the following: $$ \begin{align} ...
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2answers
51 views

Check equivalence of quadratic forms over finite fields

How to check whether the two quadratic forms \begin{equation} x_1^2 + x_2^2 \quad \text{(I)}\end{equation} and \begin{equation} 2x_1x_2 \quad \text{(II)} \end{equation} are equivalent on each of ...
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1answer
42 views

Over $F_5$, why does $f(1)=-2$ where $f(x)=x^2+2$

I am working over $F_5$ and $f(x)=x^2+1$ I am told that $f(1)=-2$. I understand that $-2=3$mod$5$ Why can we not leave it as $f(1)=1^2+2=3$? Because $3$ mod $5$ $=3$ so why do we have to "change" ...
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0answers
13 views

primitive root modulo prime powers [duplicate]

Suppose that $g$ is a primitive root modulo $p$. Show that, modulo $p^h$ for $h\geq 2$, every primitive root has the form $g'=g+np$ for a certain integer $n$. The proof of the previous statement ...
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17 views

I'm stuck on an exercise regarding finite fields and divisibility of polynomials.

We have $GF(p)$ where $p$ is some prime. The polynomial $f(x)$ is irreducible over $GF(p)$. Show that $f(x)$ divides $g(x) = x^{p^{n}}-x \in GF(p)[x]$ if and only if deg$(f(x))$ divides $n$. I assume ...
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6 views

Spinor norm of the pth power of a matrix

Let $F_{q}$ be a finite field of order $q=p^{r}$ ($p$ odd) and let $V$ be a $3$-dimensional vector space over $F_{q}$. Consider the subgroup $\Omega(3,q)$ of $SO(3,q)$., where we are picking the ...
2
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0answers
35 views

Bit operations to count longest string of 1s in a binary number - connections to FFT?

I found this rather applied question on another forum. How to calculate size of largest string of consecutive 1s in a binary number. However the other forum had neither much of a focus on applied ...
2
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2answers
31 views

Power sums over additive subgroups of finite fields

I recently read a thread on this site that solved the following problem: let $K:=\mathbb{F}_q$ be a finite field of $q$ elements and $i$ an integer. Then $\sum\limits_{\alpha \in K} \alpha^i = 0$ ...
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2answers
34 views

Zeroes of the polynomial $f(x)$ over the field $F$ of order 256.

Let $F$ be a field with 256 elements and $f \in F[x]$be a polynomial with all roots in $F$. Then (1) $f \neq x^{15} -1$. (2) $f \neq x^{63} - 1$ (3) $f \neq x^2 + x + 1$ (4) if $f$ ...
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0answers
12 views

Hardness of Solving multivariate quadratic systems

I know that solve multivariate quadratic systems over finite finite fields is a problem NP-Complete, but for instances that can be solved by computers, (e.g. using the F4 algorithm), my doubt is, ...
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63 views

Multiplicative group of $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ is cyclic

This question has been explored thoroughly, and in more generality too. For general fields, I am aware of standard proofs. However, I was naively trying to prove it in the simple case of prime $p$ ...
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0answers
15 views

Union of generators span union of subspaces

Consider a $2^k \times n$ binary matrix $C$ with the property that each $2^k \times k$ submatrix $C_i$ that contains a particular column $\textbf{c}_x$ is a permutation of all vectors in ...
2
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1answer
17 views

The connection between roots and primitive element

given a field $F=GF(q)$ and an irreducible polynom of second degree $f(x)$ over $F$ I create the extention field $F'=GF(q^2)$. given $\beta\in F'$ a root of $f(x)$, is $\beta$ is primiive element of ...
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1answer
31 views

The degree of the field extension $\mathbb{Q}(\sqrt{5},w): \mathbb{Q}$

Compute the degree of the field extension $$\mathbb{Q}(\sqrt{5},w): \mathbb{Q} ,$$ where $ w = e^{2\pi i / 3}$. I consider the tower of fields $\mathbb{Q} \subset \mathbb{Q}(\sqrt{5}) \subset ...
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1answer
44 views

Generalised Gauss sums

Let $\chi$ be a non-trivial Dirichlet character modulo an odd prime $p$ and let $f(x) \in \mathbb{Z}[x]$ be a polynomial. We define the generalised Gauss sum $$ G(\chi, f):=\sum_{y \in \mathbb{F}_p^*} ...
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1answer
47 views

Cardinality of a finite field is $p^{n}$ [duplicate]

Theorem: Let F be a finite field of characteristic p. Then p is a prime and $\left | F \right |=p^{n},\left [ \exists n>0 \right ] \in \mathbb{N}$ Note that the characteristic of F ...
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Homotopy continuations for solving systems of equations over a finite field

A way of solving systems of polynomial equations over $\mathbb{R}$ or $\mathbb{C}$ is using homotopy continuation. Roughly speaking this method uses a homotopy that starts from some system of ...
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1answer
37 views

Isomorphism between two polynomial quotient fields

What, generally, is the strategy for proving if two polynomial quotient fields are isomorphic? Say from $\mathbb Z_{11}[x]/\langle x^2+1\rangle$ to $\mathbb Z_{11}[x]/\langle x^2+x+4\rangle$? My first ...
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36 views

The Cardinality of a subset of field of 8 elements

Let $F$ be a field of $8$ elements. Let $A$ be a subset of $F$ and $A = \{x \in F \mid x^7=1$ and $x^k \neq 1$ for $k<7\}$. Then find the number of elements in $A$. Argument: Since $F$ is a ...
3
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1answer
50 views

Roots of the equation $x^2+1=0$ in $\Bbb Z/p^{n}\Bbb Z$

Let $p$ be an odd prime number and $n$ be a positive integer. I want to consider roots of the equation $x^{2}+1=0$ in the ring $\Bbb Z/p^{n}\Bbb Z$. Suppose $n=1$. Find a condition on $p$ such ...
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1answer
64 views

Is the field $\mathbb{F}_{64}$ the splitting field of $x^8-x$ over $\mathbb{F}_4$?

Is the field $\mathbb{F}_{64}$ the splitting field of $x^8-x$ over $\mathbb{F}_4$? What I find is: Let $E$ be the splitting field of $x^8-x$ over $\mathbb{F}_4.$ Thus, we have $E$ is isomorphic to ...
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3answers
48 views

Is $\mathbb{F}_{81}$ a field extension of $\mathbb{F}_{27}$?

Is $\mathbb{F}_{81}$ a field extension of $\mathbb{F}_{27}$? If it is, what is $[\mathbb{F}_{81}:\mathbb{F}_{27}]$? In this case, $\mathbb{F}_{81}$ means a field with 81 elements. I know like ...
2
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1answer
32 views

Minimum cardinality module for a fixed finite ring

Let $F$ be a finite field and $k$ be a positive integer. Let $M_k(F)$ denote the ring of $k\times k$ matrices. $M_k(F)$ is an $M_k(F)$-module with matrix multiplication, and $F^k$ is an ...
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1answer
27 views

Finite fields and generators of Galois group [closed]

I'm trying to find the order and all generators of the Galois group of $GF(256)$ over $GF(16)$? How to approach this problem?
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1answer
29 views

How to prove $t(x)-x\in F_{p}$ when $p(x) \in K$, $K$ a field whose characteristic is $p>0$

Suppose $K$ is a field whose characteristic is $p>0$, and $L$ is a finite Galois extension of $K$. Also, we have an endormorphism $h$ such that $h(x)=x^p-x$ for any $x \in L$. Question: If ...
0
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1answer
30 views

Find normal basis of the field $GF(3^6)$ and find the normal matrix

I am working with a homework is about normal basis on fields GF and I want opinions and maybe if you can help me in some doubts. 1) Find normal basis of the field $GF(3^6)$ which is understood as a ...