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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Choose a basis of $\mathbb{F}_q/\mathbb{Z}_p$ to do inverse quickly.

Let $\mathbb{F}_q$ be the finite field with $q$ elements ($q=p^n$, $p$ is a prime). $\mathbb{F}_q$ can be regarded as a linear space over the field $\mathbb{Z}_p$ of dimension $n$. The question is: ...
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How To Prove That The Rijndael Polynomial Is Irreducible?

I am learning about the AES algorithm which uses the finite field ${\mathbb{Z}_2[x]}\over{(p(x))}$, where $p(x)=x^8+x^4+x^3+x+1$. How do you prove that this polynomial is irreducible? I know that for ...
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Sums involving binomial coefficients in a finite field

Consider the field $\mathbb{F}_q$ where $q=p^k$ for some prime $p$. I have some identities related to binomial coefficients over such a field, which I wish to prove. So, can someone tell me a source ...
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37 views

Zech Logarithms

Let $a$ be a primitive element of $\mathbb{F}_{16}$ that satisfies the equation $a^4=1+a$. The logarithm of $1+a+a^2$ in $\mathbb{F}_{16}$ with base $a$ is the integer $i$ such that $0≤i<15$ ...
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Non-separable, infinite field extensions of non-zero characteristic

I have been trying to find examples (and non-examples) of fields which are separable, finite and have characteristic equal to zero. Separable Example: $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ because the ...
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Show there exists an $\alpha\in F_{p^n}$ such that $\alpha +\alpha^p+\alpha^{p^2}+\cdots+\alpha^{p^{n-1}}\neq 0$

This is part of a problem, 7.22c in "Ireland and Rosen" (self-study). In the prior problem it is shown that for prime $p$ and $\alpha\in F_{p^n}$, $f(x)=(x-\alpha)(x-\alpha^p)(x-\alpha^{p^{2}})\cdots(...
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38 views

Show that the simple extension $K(X)$ of $K$ has intermediate fields $K \varsubsetneqq F \varsubsetneqq K(X)$.

Show that the simple extension $K(X)$ of $K$ has intermediate fields $K \varsubsetneqq F \varsubsetneqq K(X)$. I have tried this: Suppose that $K \subseteq E \subseteq K(X)$. Let $\alpha \in K$ and ...
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42 views

Is there an algorithm to find the splitting field of a polynomial over galois finite fields?

how can I find the splitting field of polynomial $x^{13}+1$ over $GF(2)$?
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20 views

Finite separable fields extensions and discriminant

I am supposed to prove that for a finite separable field extension $L/K$, the discriminant $Discr_{L/K}$ is not zero. (For a basis $\{a_1,\ldots,a_n\}$ of $L$ over $K$, the discriminant is defined by ...
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Splitting field of $x^9-x$ over $\mathbb{Z}_3$.

Let $F$ be a splitting field of $x^9-x$ over $\mathbb{Z}_3$. $1$. Show that there is an $\alpha \in F - \mathbb{Z}_3$ such that $\alpha^2=2$. $2$. Show that $\mathbb{Z}_3(\alpha)$ is a splitting ...
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$\mathbb{F_2}(\alpha)/\mathbb{F_2}$ is a splitting field of $f(x)$ over $\mathbb{F_2}$

Let $\alpha$ be a zero of $f(x)=x^3+x+1 \in \mathbb{F_2}[x]$. Show that $\mathbb{F_2}(\alpha)/\mathbb{F_2}$ is a splitting field of $f(x)$ over $\mathbb{F_2}$ So we need to show that $\mathbb{...
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25 views

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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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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50 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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25 views

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

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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25 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: $$f(x)=\prod_{...
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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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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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24 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 ...
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67 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 $F(\alpha,\beta)=F(...
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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.
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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 ...
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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 \!\!\!\! \pmod ...
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40 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 ...
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42 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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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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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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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 identity)...
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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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$X^n + X + 1$ 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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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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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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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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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} f^{...
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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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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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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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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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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 ...
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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 ...
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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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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 $f$ has no ...
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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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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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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 $\mathbb{F}_2^...
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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
32 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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47 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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49 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 ...