# Tagged Questions

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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### 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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### General Multiplication Algorithm for Finite Field of GF(p^m) for small values of p^m

I am working on a small computer program to generate multiplication table of general finite field in form of GF(p^m) where p is a prime number and m is an integer greater than one. While the special ...
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### Find all the elements that are fixed by $Frob_3$

Consider the finite field $\mathbb F_9 = \mathbb F_3[x]/ \langle x^2 + 1\rangle$, and recall the Frobenius isomorphism $Frob_3 : \mathbb F_3 → \mathbb F_3$, given by $Frob_3(x) = x^3$ Find all the ...
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### Is $t^4 + 7$ irreducible over $\mathbb{Z}_{17}$?

There aren't any linear factors, $-7$ doesn't have a square root in $\mathbb{Z}_{17}$ so I've ruled out those 'easy' factorisations. I'm not sure how to continue. Looking for a hint, not a solution.
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### Finding the solution of smallest magnitude involving a non-injective matrix

Let $A$ be an $n \times n$ matrix that is non-injective, specifically one where the entries are in $\mathbb{Z}_2$ or, equivalently, $GF(2)$. Let $b$ be an $n\times 1$ matrix that is in the image of $A$...
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### Finite Fields and Discrete Log Problem

The finite field $\mathbb F_{p^p}$, for a prime $p$, can be described as $\mathbb F_p[x]/(x^p−x−1)$. Let $G=\mathbb F_{p^p}^\times/\mathbb F_{p}^\times$. Estimate the cost of obtaining the discrete ...
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### Proving existence of an element of trace 1

Let $F=\mathbb{F}_{q}$ be a finite field of order $q=2^{n}$ and let $\beta$ be a primitive element of $F$. I would like to prove that if $q>4$, then for each $1\leq i \leq \frac{q-2}{2}$, there ...
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### What are the elements of GF(9)? What is the 'addition' and 'multiplication' operations on this field?

I have read methods to construct GF(p^m). I have understood the primitive polynomials and other concepts but I have not understood how the p and m are entering into the discussion. And finally the ...
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### Is a finite field algebraic extension of a fixed degree unique or just unique up to isomorphic?

well,if $[F_{q}(\alpha):F_{q}]=m$, can i prove that $F_{q}(\alpha)=F_{q^{m}}$ ? or i can only prove that $F_{q}(\alpha)$ is isomorphic to $F_{q^{m}}$ ? thanks for advance.
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### Action on its generators of splitting field of $x^4 +5$

Splitting field is $K=\mathbb Q (\sqrt[4]{-5}, i)$ Degree of $K$ over rationals is $8$ so the galois group $G=\text{G}(K/ \mathbb Q)$ has order $8$. $x^4 +5$ is irreducible so there is one orbit ...
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### I have vectors over a finite field GF(p) where p is power of a prime. How are all the different ways in which I can define the length of a vector?

I have vectors over a finite field GF(p) where p is power of a prime. What are all the different ways in which I can define the length of a vector?
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### Want to know the best books on Matrix algebra dealing with the special case of over GF(2).

Want to know the best books on Matrix algebra dealing over GF(2). Want to know the special properties of matrices over GF(2). For example, in GF(2), non-zero vectors can have a dotproduct with ...
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### Proving that $x^4+x^3+1$ has no solution over $\mathrm{GF}(2^e)$, $e$ odd.

I'm trying to find out whether $x^4+x^3+1$ has a solution over $\mathrm{GF}(2^e)$, $e$ odd. Some quick calculations for small values of $e$ indicate that it does not, but my background is not in ...
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### Plotting Elliptic Curve over Finite Field in Maple

Not sure if I might be in the wrong section, but I am looking for guidance on how to plot an elliptic curve over a finite field in Maple. I have tried looking it up but only getting good results for ...
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### Easy way to find the order of elements in a finite field

I am trying to work out the multiplicative order of each non zero element in $F_7$. Lets say I am looking at the number $3$. I know its order is $6$. Instead of having to work out the powers of ...
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### Finding the spinor norm of an element using a proposition in 'The Maximal Subgroups of the Low-Dimensional Finite Classical Groups'.

Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an ...
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### Show irreducible polynomial by factorization in $\mathbb{F}_2[X]$ and $\mathbb{F}_3[X]$

I found a problem, I don't really know how to solve, although it should be something very easy, since it is stuff of Algebra I. Let $f= 29X^5−13X^4−44X^3+ 18X^2+ 35X+ 10\in\mathbb{Z}[X]$. 1) ...
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### Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$

Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$. Work so far: $$x^8 - x = x(x^7 - 1) = x(x - 1)(x^6+x^5+x^4+x^3+x^2+x+1)$$ From here, I think the Zeros of an Irreducible over a ...
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### Discrete math: find prime number which solves the following conditions

Find prime number $p$ and polynomial $f$, so that the ring $F_p[x]/(f)$: 1) contains non-zero element $w$: $w^n=0$ for some $n$ 2) doesn't satisfy 1), but is a field 3) is a field which contains $8$...
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### How to find the elements of a finite field?

OK, I am asked to find the elements for the addition and multiplication tables for the finite fields with eight elements. I know I've asked about this question previously, and I've almost gotten a ...
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### Cyclic code and its generating polynomial

Can someone give me an example of a $[3,2]$ linear cyclic code and its generating polynomial? Does this mean it is a binary cyclic code with length 3? Also how do I find all binary/ternary cyclic ...
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### Kloosterman Sums and K3 Surfaces

In R. Livne's Motivic Orthogonal Two-Dimensional Representations of $\mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, he provides a technique for computing the $\zeta$-function of the $K3$ surface X=\{...