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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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 ...
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1answer
26 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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1answer
17 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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2answers
26 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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22 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 ...
2
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26 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
23 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 ...
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0answers
9 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 $\ell$ is a prime and $\gcd(\ell, q)=1$. I have the following question: Does there exist an element ...
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39 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 ...
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2answers
20 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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27 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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0answers
18 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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1answer
41 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
30 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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33 views

Primitive element in multivariate Galois field [on hold]

On Singular CAS I can define a Galois field $(2^3)$ with $(x,y,z)$ variables. But I am not able to understand how $a^3+a+1$ is still its primitive element. General example taken in books is always ...
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1answer
49 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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1answer
52 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 ...
2
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2answers
32 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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37 views

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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1answer
26 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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0answers
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 $$[0] = 0 + I$$ $$[1] = 1 + I$$ $$[i] = i + I$$ ...
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1answer
33 views

Proving that if the cardinality of a field $F$ is finite and equal to $q$, then the ring $F[X]/(X^n)$ is finite of cardinality $q^n$

I'm trying to prove how the cardinality of a field $F$ is finite and equal to $q$, then the quotient ring $F[X]/(X^n)$ is finite of cardinality $q^n$. How do I go about this when the quotient ...
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0answers
23 views

Zeroes and extension fields [closed]

I am having trouble with this question (i.e., I have no idea where to begin). $p$ is a prime number and $E$ is an extension field of $\mathbb Z_{p}$ (integers mod $p$). Let $f(X)∈\mathbb ...
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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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3answers
3k views

Irreducible polynomial which is reducible modulo every prime

How to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$? For example I know that $x^4+1=(x+1)^4\bmod 2$. Also $\bmod 3$ we have that $0,1,2$ are not ...
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5answers
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Do we really need polynomials (In contrast to polynomial functions)?

In the following I'm going to call a polynomial expression an element of a suitable algebraic structure (for example a ring, since it has an addition and a multiplication) that has the form ...
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2answers
50 views

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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20 views

$f(x) = x^2 + bx + a$ 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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1answer
56 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 ...
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16 views

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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1answer
37 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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17 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
23 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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2answers
61 views

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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2answers
46 views

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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16 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) ...
4
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3answers
85 views

Reducibility of polynomials modulo p [duplicate]

How do we show that $X^4-10X^2+1$ is reducible modulo every prime $p$? I've managed to show it for all primes less than 10, for primes greater than 10 we have $X^4+(p-10)X^2+1$. Where do I go ...
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61 views

Does NTT have an upper bound?

I'm working with NTT (Number theoretic transform) to reduce the complexity of a polynomial multiplication. For this, I'm using $P = 2^{64}-2^{32}+1$ to generate the primitive root $\omega_N$ needed by ...
3
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36 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} ...
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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 ...
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1answer
18 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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77 views

Find the number of primitive elements

How can i find the number of primitive elements over the field of order q? GF(27) for example. Is there a formula that I can follow? I'm really confused on how to find them. Any help would be much ...
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2answers
19 views

Using linear algebra operations over a finite field in MuPad

When I try to use MuPad's linear algebra operations over a finite field, I receive the error 'An arithmetical expression is expected.' How should I go about doing this? ...
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25 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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92 views

Solutions of affine polynomials in characteristic $2$

I will give an example for expressing where I stuck about affine polynomials: Let $\alpha \in \mathbb F_{2^k}^*$. Let $L_{\alpha}(x)=x^4+x^2+\alpha x$ be a linearized polynomial over $\mathbb ...
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1answer
69 views

Is the numbers of primes that is sum of 2 + another prime is finite?

In order to have sum of $2$-primes to be a prime one of the primes must be the prime $2$. However the "distance" between adjacent primes increases as we search along the natural numbers. For example ...
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1answer
19 views

$F$ be a finite field , then are there infinitely many polynomials $f(x) \in F[x]$ such that $f(a)=0 , \forall a \in F$ ?

Let $F$ be a finite field , then is it true that there are infinitely many polynomials $f(x) \in F[x]$ such that $f(a)=0 , \forall a \in F$ ?
2
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1answer
16 views

Potential Frobenius automorphism question

Let $F$ be a finite field of characteristic $p$ of size $p^n$ for $n \ge 1$ with the base field $K \cong Z_p$. I'm attempting to prove that the map $\phi: F → F$ sending $u$ to $u^p$ for each $u \in ...
2
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1answer
24 views

Theorem on Repeating Decimals

So I am wondering if anyone recognizes the following theorem: Given a prime $p$, and a base $b$ (natural number $>1$), the period of $\frac{1}{p}$ expressed in base $b$ is the unique $d$ that ...
2
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2answers
26 views

relationship between monomorphism and automorphism [duplicate]

If I have a monomorphism $\phi : L \to M$ where $L,M$ are fields with $L \subset M$ and I know that $\phi(L) \subset L$ does that necessairly mean that $\phi : L \to L$ is an automorphism? I can ...