# Questions tagged [finite-fields]

Finite fields are fields (number systems with addition, subtraction, multiplication, and division) with only finitely many elements. They arise in abstract algebra, number theory, and cryptography. 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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### Irreducibility of $t ^ { 4 } + t - 1$ over $\mathbb{F}_{25}$

Problem: Let $f ( t ) = t ^ { 4 } + t - 1 \in \mathbb { F } _ { 5 } [ t ]$, show that $f(t)$ has no roots in $\mathbb{F}_{25}$. I tried the following: suppose $f(t)$ has a root in $\mathbb{F}_{25}$, ...
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### Must Additive Inverse elements be opposite in sign?

Is it possible to define a field $F=\left \{ 0, 1, a \right \}$ where the Additive Inverse condition is expressed as : $x+x=0 \space \space \forall x\in F$ ? My doubt comes from reading my book on ...
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### Create projective plane

Please explain this https://math.stackexchange.com/a/463369/672948 in a simpler way. I am not from higher mathematics background and these terms are quite hard to understand. I am clear upto finding ...
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### Choosing a zero-sum sequence from a finite field. [duplicate]

Let $\mathbb{F}_p$ be a finite field (for a prime integer $p$), and let $x_1, \ldots, x_{2p-1}$ be any sequence of elements from it. Prove that I can chose $p$ elements from this sequence such as ...
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### Proving an isomorphism of Galois group

Let $p$ be a prime number and $\alpha\in\mathbb{N}$ such that $\forall\beta\in\mathbb{Q} \space\alpha\neq \beta^p$ e.g. $\alpha$ is not a $p$-th power of any rational number. Let $E$ denote the ...
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### Elliptic Curve over a Finite Field, Adding Graphically

I use Mathematica to add two points graphically on the elliptic curve $y^2 = x^3 + 3x + 8$ over $\mathbb{F}_{13}$. Specifically, I'd like to illustrate $(1,8)+(2,10)=(1,5)=(1,-8)$, but on first glance,...
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### Subrings $A$ of $\mathbb{F}_p[x]$ such that $\dim_{\mathbb{F}_p}\mathbb{F}_p[x]/A=1$.

Suppose we have the ring $\mathbb{F}_p[x]$ of polynomials with coefficients in the Galois field with $p$ prime number elements. I want to find all subrings $A$ containing the identity such that when ...
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### Degree of a splitting field of an irreducible polynomial over $\mathbb{F}_p$

Is it true that whenever $p$ is an odd prime, and $f$ an irreducible polynomial of degree $p$ in $\mathbb{F}_p$, then the splitting field of $f$, denoted $L$, satisfies $[L:\mathbb{F}_p] = p!$ ? I ...
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### Multivariate variate polynomials over finite field with common root will have another root, probabilistic proof?

I have a finite field $F_p$ and $m$ polynomials $P_1, P_2, \dots, P_m$ in this field. Also every polynomial is multivariable, so $P_i$ from $F_p^n$ to $F_p.$ It is known that $n>\sum_i \deg(P_i).$ ...
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### how many points belong to the quadric $x_0^2+x_1^2+x_2^2+x_3^2=0$ in $\mathbb{P}_3$ over $\mathbb{F}_9$

I have a problem with the following question: how many points belong to the quadric $x_0^2+x_1^2+x_2^2+x_3^2=0$ in $\mathbb{P}_3$ over $\mathbb{F}_9$. How I tried to solve this problem. Here we have ...
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### If $K$ is an extension of $\mathbb{Z}/p\mathbb{Z}$, every elements of $K$ is a root of $t^{p^n -1}-1$

Problem : Show that the characteristic of a finite field $K$ is a prime number. Show that $K$ is an extension of $\mathbb{Z}/p\mathbb{Z}$ and there exists $n$ such that Card($K$) $= p^n$. ...
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### Conjugacy of Singer cyclic groups in $\mathrm{P\Gamma L}$

Motivation This is kind of a follow-up to this question on conjugacy of Singer cyclic groups in GL. The "original" definition of a Singer cycle is not in the GL, but the following slightly different ...
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### trace of Frobenius

how can I calculate trace of Frobenius for a single point on an elliptic curve $E(F_{q^{12}})$? I've tried to sum up 12 points that were different powers Frobenius maps but none of the points don't ...
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### additive order of any zero divisor in $Z_{p^2}$ is p, is it true?

This result was used in a proof of a theorem, i am not sure if it's true. can someone tell the proof idea. Can it be generalized to additive order of any zero divisor in $Z_{p^k}$, is there any ...
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### Let $F$ be a finite field with $\text{char}(F) = p$. Now, if $u$ is a primitive element, show that $u^p$ is also primitive.

I need help in understanding how to prove this. I know that if $u$ is a primitive element of a Finite field, $F$, then $u$ generates $F^*$.
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### Interpolation of a rational function

Assume I am given two polynomials $f(x)$ and $g(x)$ with coefficients from a field $\mathbb{F}_p$, where $p$ is a prime. Now I know that the set of these polynomials is a ring and not a field, meaning ...
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### $\mathbb{F}_p({\rm i}) = \mathbb{F}_{p^2}$ iff $p=3\mod 4$

I know the fact that \begin{equation*} \mathbb{F}_{p^2} = \mathbb{F}_p({\rm i}) \simeq \mathbb{F}_p[x]/(x^2+1) \end{equation*} for a unit ${\rm i}$, i.e. ${\rm i}^2=-1$, iff $p=3\ {\rm mod}\ 4$ for a ...
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### How many solutions of the equation $ax^2 +by^2 = 1$ are there with $(x, y) ∈ \mathbb{F}_{p} ×\mathbb{F}_{p}$ [duplicate]

How many solutions of the equation $ax^2 +by^2 = 1$ are there with $(x, y) ∈ \mathbb{F}_{p} ×\mathbb{F}_{p}$ where $a, b$ are integers whose product is not divisible by $p$? This was a recommended ...
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### How many elements of have square roots in a field of 13 elements? [duplicate]

Initially, I thought since this field was isomorphic to $({0,...,12})$ , the elements $(0,4,9)$ would have square roots. However, when I checked the solutions, the answer was different. Thank you in ...
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### proving $x$ is the generator of a cyclic group

Show that $x$ is a generator of $(\mathbb{Z}_3[x]/\langle x^3+2x+1\rangle)^*$. I don't understand part of the solution. $x^3+2x+1$ is irreducible in $\mathbb{Z}_3$. Let $a$ be a zero of $x^3+2x+1$ in ...
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