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 $\operatorname{GF}(q)$.

learn more… | top users | synonyms (1)

0
votes
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?
1
vote
2answers
43 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 ...
4
votes
0answers
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 ...
0
votes
2answers
32 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?...
2
votes
1answer
50 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, ...
0
votes
0answers
79 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 $\...
1
vote
0answers
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\...
1
vote
2answers
36 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 ...
2
votes
0answers
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&...
2
votes
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. ...
0
votes
0answers
22 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 ...
1
vote
1answer
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 ...
1
vote
1answer
36 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 ...
1
vote
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 ...
2
votes
2answers
57 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 ...
3
votes
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 ...
2
votes
1answer
30 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 ...
0
votes
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 $$[0] = 0 + I$$ $$[1] = 1 + I$$ $$[i] = i + I$$ $$[i+...
-2
votes
2answers
45 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 $E=\mathbb Z_5[X]/(f(X))$. ...
0
votes
0answers
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 ...
1
vote
0answers
23 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})} ...
1
vote
2answers
54 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 ...
4
votes
0answers
17 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 ...
1
vote
1answer
42 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?
1
vote
0answers
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 $$\...
0
votes
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 ...
0
votes
2answers
56 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 ...
1
vote
2answers
68 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 ...
0
votes
0answers
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) \...
3
votes
0answers
21 views

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,...
1
vote
1answer
20 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 ...
0
votes
0answers
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 ...
2
votes
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
votes
1answer
97 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} \...
1
vote
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 ...
1
vote
1answer
20 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
votes
1answer
22 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 $u \in F$...
3
votes
1answer
33 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 ...
1
vote
2answers
32 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? ...
2
votes
2answers
30 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 ...
0
votes
0answers
36 views

existence of irreducible polynomial of degree 10 over finite field

Prove that there exists an irreducible polynomial of degree 10 over the the field of 25 elements. I know that the multiplicative group of non-zero elements of any finite field is cyclic. So how can I ...
0
votes
1answer
53 views

How do I find the ideals in the ring $\mathbb F_3[x]/(x^2+2)$?

Clearly $\{0\}$ and $\mathbb F_3[x]/(x^2+2)$ will be ideals. How would I find the others?
2
votes
1answer
54 views

Any method to solve this system of equation?

We have m variables $ x_{1},x_{2},...,x_{m} $ which are elements of field $F_{p}$ and we are given m equations of the form $$\sum_{i=1}^{m} x_{i}^{n} = c_{n} \mod p \qquad for \: 1 \le n \le m$$ ...
0
votes
1answer
49 views

Is $K^{n}$ Zariski Hausdorff when $K$ is a finite field?

Assume that $K$ is a finite field. Is it true to say that $K^{n}$ is a Hausdorff topological space with Zariski topology?
0
votes
0answers
24 views

I am currently working on the implementation of the Powerline System which is based on the Chor-Ri

I am currently working on the implementation of the 'Powerline System' which is based on the Chor-Rivest cryptosystem for my number theory project. There is a step in the key-generation phase of the ...
-1
votes
1answer
38 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 $q^n$ [closed]

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 ring ...
1
vote
0answers
11 views

Transforming a polynomial sum using a series expansion (BCH codes)

In my study of BCH codes I've come across the following equation (the "key equation"): $$ \Omega(x) \equiv \Lambda(x)S(x) \mod x^{n-k} \tag{1} $$ Where the two terms on the right are defined by: $$ ...
2
votes
0answers
69 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 ...
11
votes
2answers
64 views

Show that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $

I read in a paper that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $ this is counterintuitive / surprising since $\mathrm{SO}_3(\mathbb{R}) \not \simeq \mathrm{SL}_2(\mathbb{R}) $ ...
0
votes
2answers
54 views

Totien-Sum: why GCD( {n}/d, q/d) = 1; implies Sum{Totient(d/q) } = q

Have seen answer to this question. still don't understand.. Totient sum is defined: q = Sum(Totient (d) ); sum on all d : d|q More specific; The proof has these steps: 1. If d is a divider ...