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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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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19 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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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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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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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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40 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.
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45 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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65 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 \!\!\!\! ...
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1answer
27 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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39 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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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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How to attack universal hash function based on finite-field arithmetic?

As per the Recursive n-gram hashing is pairwise independent, at best paper, I want to use the algorithm described in chapter 6 and 7 (page 7 - 10). The hash works as follows: Define a random ...
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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
103 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
62 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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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
26 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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52 views

I don't understand a step of the proof that a nonzero finite commutative ring with no zero divisors is a field.

This seems to be a fairly classic problem in algebra: Let $R$ be a nonzero finite commutative ring with no zero divisors. Prove that $R$ is a field. Here is the a solution I came across: ...
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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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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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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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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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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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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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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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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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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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Give $3$ examples of a field extensions which are neither normal nor separable.

Here $1)$ $F=\Bbb F_2(x)$ and consider $K=F(x^{1/6})$. Now $K/F$ is neither normal nor separable. $2)$ Let $k$ be field of characteristic $2$, let $F=k(x,y)$, let $S=F(u)$, where $u$ is a root of ...
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Extending a finite field twice

Assume we have a finite field $\mathbb F_p$, an irreducible polynomial $f(x)$ of degree $m$ over $\mathbb F_p$, and an irreducible polynomial $g(y)$ of degree $n$ over $\mathbb F_p[x]/(f(x))$. Then ...
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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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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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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 ...
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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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29 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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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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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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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 ...
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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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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 ...
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31 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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28 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 ...
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51 views

13th root of 2 in field $\mathbb{F}_{13}$ [on hold]

Is there an easy to find $13^{th}$ root of $2$ in the field $\mathbb{F}_{13}$? I'm having trouble finding one here. Thanks!
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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 ...
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Order of an orbit of Frobenius action on a algebraically closed field of characteristic p

Consider the action of the Frobenius homomorphism $F^{2}:\,\overline{\mathbb{F}_{q}}\rightarrow\overline{\mathbb{F}_{q}},\,x\rightarrow x^{q^{2}}$ over $\overline{\mathbb{F}_{q}}$ . Let $s=\left\{ ...
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10 views

irreducibility test for multinomials over a finite field

I am working in an algebraic cryptosystem, and I need, in the process, ensure that a 3-variables polynomial in a finite field is irreducible but I can't find a practical method to do that. Do you know ...
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1answer
16 views

How are Gauss sums the analogue of the gamma function for finite fields?

I've seen this statement on the internet in a few places and I don't really get the connection. Would anyone mind fleshing out the details? Thanks.