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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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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52 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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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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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 $\mathbb{...
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1answer
35 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 $M_k(F)$-...
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1answer
28 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
31 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 $t\...
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1answer
33 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 ...
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1answer
26 views

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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1answer
25 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
20 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.
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23 views

What does "class $x \in X$ mean?

I'm working through Escofier's book on Galois Theory, and in several exercises regarding finite fields they use the terminology "class $x \in X$, and I'm not certain what it means. For instance, let $...
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2answers
55 views

Subfield of a finite field

I have started studying field theory and i have a question.somewhere i saw that a finite field with $p^m $ elements has a subfield of order $p^m $ where $m$ is a divisor of $n $.My question that if ...
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43 views

Finite field and its element with symbols [Sage / Python / …]

I have a finite field $T=GF(2^3)$, normal basis $(a, a^2, a^4)$ and polynomial $f$ from field $T$, which contains unknown variables / symbols. Is it possible to get vector with coordinates of f in ...
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1answer
72 views

Factor $x^6+x^5+x^4+x^3+x^2+x+1$ in $\mathbb{F}_2[x]$ [duplicate]

I'm trying to factor $x^6+x^5+x^4+x^3+x^2+x+1$ in $\mathbb{F}_2[x]$. But I don't know how to do that. Anyone can tell whether there is a nice way to solve all these kinds of problems?
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17 views

Primitive polynomials from GF(q) to GF(q^n)

Suppose that over some finite field $GF(q)$, we have two monic primitive polynomials of orders $n$ and $mn$. -From these polynomials, is there always a 'natural' monic primitive polynomial over $GF(q^...
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1answer
30 views

The relation between $GF(2)$ and $GF(2^3)$

Both $GF(2)$ and $GF(2^3)$ are finite fields of characteristic $2$. Is $GL(2^3)$ an extension of $GF(2)$? Can someone point some links that details something about this, please?
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2answers
40 views

How to show that $X^p-t\in\mathbb{F}_p(t)[x]$ is irreducible? [duplicate]

This question is previously asked here, but there is no complete solution of it. I understand that the root $\alpha$ exist in the algebraic closure of $\mathbb{F}_p(t)[x]$, and it is the only root ...
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99 views

Number of $\mathbb{F}_q$-rational points on a smooth variety

From the proof of Weil's conjectures it follows that $|q^k - \# X(\mathbb{F}_{q^k})| = O(q^{k(n - \frac{1}{2})})$, where $X$ is a smooth variety over $\mathbb{F}_q$ and $n = \dim X$ (see for example ...
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2answers
31 views

Bruhat decomposition and the order of the group.

Suppose we have a group $G(q)$ over a finite field $\mathbb{F}_q$. How can the Bruhat decomposition be used in order to calculate the order of $G(q)$? Are there any examples for some particular groups?...
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1answer
49 views

Additive combinatorics and $|S \cap (S+t)| \geq K |S|$ over $\mathbb{F}_2^n$

I don't know much about additive combinatorics, and I am wondering if there are results concerning the size of $|S \cap (S+t)|$ where $S \subset \mathbb{F}_2^n$ and $t \in \mathbb{F}_2^n$. Especially, ...
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57 views

Trace 0 and Norm 1 elements in Finite fields

Let $F_q$ and $F_{q^{\ell}}$ be the finite fields with $q, q^{\ell}$ elements respectively, where $\ell \ge 3$ is a prime and $\gcd(\ell, q)=1$. I have the following question: Does there exist $\...
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41 views

Find the value of $n$

Let $F$ be a field having $5^n$ elements .Also $F$ has an element which satisfies $x^{5^n}=1$ such that $x\neq 1$. Find $n$ . My try: Let $x\in F $ satisfy $x^{5^n}=1$ .Obviously the group $F^{*}=F\...
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30 views

Irreducible polynomials over $GF(4)$.

I'm working with $GF(4)$ and I'm looking for irreducible polynomials of different degrees over that field. So $GF(4) = \{0,1,\alpha, \alpha + 1\}$ where $\alpha^{2} + \alpha + 1 = 0$, and first I'm ...
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24 views

Closed formula involving $q$-binomials

I was working on a combinatorial problem over finite fields, and the following quantity came up $$ \sum_{r=0}^k r\binom{n-k}{r}_q\binom{k}{k-r}_qq^{r^2},$$ where $k,n$ are integers such that $0<k&...
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2answers
29 views

How to prove that $GF(p^n)$ contains an element of degree n?

I am reading Gallian's Algebra book, and I am lapsing on Corollary 2 to Theorem 22.2. The statement goes: "Let $a$ be a generator of the group of nonzero elements of $GF(p^n)$ under multiplication. ...
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20 views

In a binary code, all coordinates partake in at least one non-information set

It is true that all non-MDS $(n,k)$ codes contains at least one $k$-sized coordinate subset that does not correspond to an information set (because all such subsets are information sets iff the code ...
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46 views

Constructing an irreducible polynomial of degree $2$ over $\mathbb{F}_p$

I want to construct an irreducible polynomial of degree $2$ over $\mathbb{F}_p$ where $p$ is a prime that can be written as $4k+1$. My attempt is as follow: we can assume that this polynomial is of ...
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1answer
35 views

Finding exact isomorphism between finite fields given as quotient rings [duplicate]

I have two quotient rings over $\Bbb F := GF(3)$: $$\Bbb F[x] / (x^3 -x - 1) \qquad \text{and} \qquad \Bbb F[x] / (x^3 -x + 1) .$$ These things I know: Both quotient rings are irreducible, that means ...
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1answer
57 views

Expected length of spanning sequences of vectors

Given an infinitely-long sequence $$L = V \cdot V \cdot V \cdots$$ ...that is the repeated concatenation of $$V = (v_1, \dots, v_{2^n - 1}), v_i \in \Bbb{F}_2^n$$ ...a sequence of $2^n - 1$ vectors ...
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2answers
52 views

Confusion about elements in fields, like -1 in Z5

I'm learning field and ring theory, and I've repeatedly seen the usage of -1, -2 and -3 as elements of $\mathbb{Z}_5$. As far as my knowledge goes, $\mathbb{Z}_5$ consists of {0,1,2,3,4}. This is ...
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1answer
55 views

Subsequences of length $n$ always making a basis for $\mathbb{F}^n_2$?

Is it always possible to order every vector of $\mathbb{F}^n_2$ (except the zero vector) as a sequence $V = (v_1, \dots, v_{2^n - 1}) | v_i \in \mathbb{F}^n_2 \setminus \{0\} $ such that every ...
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1answer
28 views

Generating spanning sets for $\mathbb{F}^n_2$?

What is the maximum value $m$ such that an unordered set of $n + m$ vectors spans $\mathbb{F}^n_2$ when any $m$ vectors are excluded? Also, is there an efficient method for generating such a sequence ...
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33 views

Quotient ring of complex numbers [duplicate]

We have a set $Z[i] = \{ a + bi \}$ and an ideal $I = ( 2+ 2i)$ I want to find a quotient ring $Z[i]/(2+2i)$ . I think the result should be $$[0] = 0 + I$$ $$[1] = 1 + I$$ $$[i] = i + I$$ $$[i+...
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Extension fields, and their cardinality and roots

I have no idea how to begin answering this question. My notes do not help. Let $f(X) = X^3$ + $X + 1 ∈ \mathbb Z_5[X]$ ($\mathbb Z_5$ denotes the integers mod 5). Let $E=\mathbb Z_5[X]/(f(X))$. ...
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22 views

Order of a map $\Bbb F_{p^n}\to \Bbb F_{p^n}$which maps $x$ to $x^{p} -x$ [duplicate]

Let $\Bbb F_{p^{n}}$ be the field of order $p^{n}$. Define a map $\phi: \Bbb F_{p^{n}}\to\Bbb F_{p^{n}}$ by $x \mapsto x^{p} -x$. My question is what is the order of im$(\phi)$? I already know ...
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$f(x) = x^2 + bx + a$ irreducible over $\Bbb F_p$ (finite field of $p$ prime elements) iff $(b^2 - 4a)^{\frac{p-1}{2}} = -1$ in $\Bbb F_p$

My attempt started as follows. I know that for $f$ to be irreducible, $D = b^2 - 4a$ is not a square in $\Bbb F_p$ (ie $(\frac{D}{p}) = -1$). I also know that $D^{p-1} = 1$, so I see $\sqrt{(D^{p-1})} ...
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Elliptic curves with trivial Mordell–Weil group over certain fields.

I am looking for elliptic curves $E,E'$ defined over $\mathbb{F}_{3}$ and $\mathbb{F}_{4}$ respectively and given by a Weierstrass equation such that their Mordell-Weil group is trivial, i.e. such ...
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Computing Multiplicative Character Values over Finite Fields [duplicate]

Let $\mathbb F_q$ be the finite field of order $q$, where $q\equiv 1\pmod 4$ is some prime power. Let $\chi_4\colon\mathbb F_q^\times\to\mathbb C^\times$ be a multiplicative character of exact order 4 ...
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40 views

Geometric intuition for finite vector spaces?

There is a powerful geometric intuition for real vector spaces. Is there any good way of visualizing vector spaces over finite fields?
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19 views

Basis of the special form

Let $R = \mathrm{GF}(q)$, $S = \mathrm{GF}(q^n), \ n\geq 2$ be extension of $R$, $h$ be a primitive element of $S$. I want to count or estimate the number $N$ of basis of the following form. Let $$\...
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1answer
24 views

Product field of intermediate fields is field extension

Let $A\subset Z$ be some field extension, and $L$ and $E$ intermediate fields of this extension. Suppose that $A\subset L$ is a finite Galois extension. 1 How do I prove that $EL$ is a finite ...
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Irreducible implies Separable in a Finite Field

Proposition 37 on page 549 of Abstract Algebra, 3rd Ed. by Dummit and Foote claims that irreducible implies separable over a finite field. Suppose $p(x)$ is irreducible over a finite field of ...
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Galois group is isomorphic to group of characters. What can we say then?

Let $\overline{K}/K$ denote the separable closure of a finite field $K$ of characteristic $p$ and let $\mu_{n}$ denote the group of $n$-th roots of unity in $\overline{K}$, where $(n,p)=1$. Let us ...
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19 views

Frobenius Map and Subfields of $\bar{\mathbb{F}}(x,y)$

Let $L=\bar{\mathbb{F}}_{p}(x,y)$ (for $\bar{\mathbb{F}_{p}}$ an algebraic closure of the finite field) and let $L^{(p)}$ be the image of $L$ under the Frobenius map $L \rightarrow L $, where $g(x) \...
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finite fields: efficient primitive element test?

Suppose $x \in F_n^{*}$, where $F_n$ is a finite field. Is there an efficient way to test whether x is a primitive element? This is the best I can come up with: You factor n-1 into all of its factors,...
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1answer
19 views

Is this a hyperplane or a half-space in $\mathbb{F}_2^n$?

Simple terminological question: the equation $x_1+\dots+x_n = 0$ over $\mathbb{F}_2^n$ is called a subspace. I'm wondering if we could also call it a hyperplane, a half-space or neither? The equality ...
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27 views

Let $F$ be a finite field, $E$ a field extension, and $A,B\in M_n (F)$ conjugate in $M_n (E)$

Prove that $A$ and $B$ are conjugate matrices in $M_n (F)$. The question is from K. Conrad's notes on Potential Diagonalizability of Linear Operators, which he proves for the case of infinite fields ...
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1answer
58 views

Solving an equation involving the trace of a field

Let $F$ be a finite field of order $q$ where $q=2^{n}$ and fix $l\in F\setminus{0}$ with $Tr(l)=0$. I want to determine the number of $a$ such that $$Tr(la)=Tr(la^{-1})=1,$$ where $Tr$ denotes the ...
4
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1answer
95 views

An equation over a finite field

Suppose $x,y,z,w \in \mathbb{F}_{q^2}$, where $q=p^k$ for some prime $p$. Consider the system of equations $$ \left\{ \begin{array}{l} xy + zw = 0; \\ xy^q + yx^q + zw^q + wz^q = 0. \end{array} \...