# 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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### Number of Solutions of the Hyperbola Equations over Finite Fields

I have a problem with proving the number of points of the hyperbola equation $H_a: x^2 + y^2 = a$ (for every a > 0 in finite field $F_p$) in the finite fields. I have to prove that the number of ...
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### Is spliting field of $f(x) \in F[x]$ with $F$ a finite field also a finite field?

Let $F$ be a finite field, and $f(x) \in F[x]$ is some non constant polynomial, Let $E$ be the splitting field of $f(x)$ over $F$ , is $E$ always a finite field? My attempt let $\deg(f) = n$ then ...
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### Multiplication of two random matrices over a finite field

Consider a matrix $\mathrm{X}$ sampled uniformly at random from the set of all rank $r$ matrices over $\mathbb{F}_q^{m \times n}$ and a matrix $\mathrm{Y}$ sampled uniformly at random from the set of ...
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### How to prove that Quaternion's algebra over isomorphic to Mat2(Z [duplicate]

My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when p = 1 (mod 4), and for p = 1 (mod 4) it get it. In my second attempt, I tried to look at them as vector ...
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### Proving that the following equation does not have integer solutions

I want to prove that the following equation has no integer solutions $a,b,c$: $$-a^3 - b^3 - c^3 + ab^2 - ac^2 + bc^2 - 2a^2c + 3abc = 0$$ apart from the naive solution $a=b=c=0$. The context, in case ...
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### Showing that $x^3 - t$ is irreducible over $\mathbb{F}_3(t)$

I was reading the post Is $\mathbb{F}_3(t,t^{1/3})/\mathbb{F}_3(t)$ a normal extension? Is it separable? I do not understand, why we can use Eisenstein's criterion to show that $x^3 - t$ is ...
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### show $ord(-1)=1$ in $K^*$ where $|K|=p^n$ is a field

Consider a finite field $K$ of order $|K|=p^n$. Denote $K^*$ the group of units. Show that if $p=2$ then the $ord(-1)=1$ in $K^*$ Show that if $p\geq3$ then $ord(-1)=2$ in $K^*$ How do I approach ...
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Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. (What he calls a 'pure extension' is commonly called 'radical extension' by most authors.) I am confused by ...
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Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. It is about the characterization of the Galois group of pure extensions (which are mostly called radical ...
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### find all irreducible polynomials of degree 2 and 3 over Z5

I would like to find all irreducible polynomials of degree 2 and 3 with coefficients in Z5. I know that the polynomial (x^5)^n - x equals with the product of all monic irreducible polynomials of ...
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### Over a finite field, which square matrices produce a zero quadratic form?

For which matrices $A \in (\mathbb{F}_p)^{n \times n}$ do we have $x^T A x=0$ for all $x \in (\mathbb{F}_p)^n$? Obviously, this is the case if $A=B-B^T$ for some $B$ (which is equivalent to saying ...
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### How to factor $X^4+5X^3-2X^2-2$ into its irreducible form over $\Bbb{Z}_{11}$ [closed]

The polynomial $X^4+5X^3-2X^2-2$ has no roots in $\Bbb{Z}_{11}$ so I am unsure as to how I am meant to factorise in such a scenario when I cannot use the factor theorem. How am I meant to progress? ...
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### When is the trace of a matrix group surjective over $\mathbb{F}_p$?

Let $p$ be a prime and $G\subset\operatorname{GL}_n(\mathbb{F}_p)$ be a subgroup. I wondering about the following question: Is the map $\operatorname{Tr}:G\to\mathbb{F}_p$ surjective? I know it's true ...
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### Is it possible to produce identically-behaving binary extension fields using different irreducible polynomials?

Let $GF(2^m)$ be a binary extension field with constructing polynomial $f(z)$ be an irreducible, primitive polynomial over $GF(2)$. Is there any possibility that two (or more) different $f(z)$ can ...
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### Polynomial in finite field where all the elements of the field are roots

Find a monic polynomial of degree n where n is a power of a prime p and every element of F_n is a root. I attempted to solve this problem by using the fact that every element in a finite field can be ...
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### Vector spaces over certain finite fields are not equal to any unions of their subspaces

From the 3rd edition of the book "The Linear Algebra a Beginning Graduate Student Ought to Know" by Jonathan S. Golan, we find the following exercise under chapter 3: : First of all, let's ...
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### Linear algebra over finite fields

Let $\mathbb{F}_p^3$ be a $3$-dimensional vector space over $\mathbb{F}_p$ with $p$ odd. For any $\mathbf{x}\in \mathbb{F}_p^3$ define its "norm" $\lVert \mathbf{x}\rVert=x_1^2+x_2^2+x_3^2,$ ...
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### A question related to the divisibility of a number

I am currently reading upon the following proof of the following statement from Sharifi's notes on Algebraic Number Theory The maximal unramified extension $K_{nr}$ of a local field $K$ is given by ...
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### DFT, but with large values available

Suppose that we are calculating a size $N$ integer-valued DFT, with some values possibly adjoined to the integers, such as the imaginary $i$. My question is, if the word size allows integers much ...
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### Equivalent of floor division in a group of integers mod N.

I'm working on a small programming project, and I'm struggling a bit with calculating fractions of numbers in a commutative group. I'm by no means a mathematician or a programmer, so please bear with ...
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### $\exists$ $a, b, \in \mathbb{F}$ such that $a^2 + b^2 = 2ab = 0$ $\implies$ Char($\mathbb{F}$) $= 2$.

Consider a field $\mathbb{F}$. If there exist elements $a, b \in \mathbb{F}$, not both zero, such that $a^2 + b^2 = 0$ and $2ab = 0$, then I need to show that the characteristic of $\mathbb{F}$ is $2$....
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### Show that the units of a finite field form a cyclic group.

I have shown that for a finite commutative group, there is an element $x$ such that the order of every other element divindes the order of $x$. I was thinking how I could apply this to prove that the ...
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### Representation theory over finite fields?

This is mostly a reference request ... I think. I am a bit familiar with representation theory of finite groups. Here I have seen a representation as a homomorphism $\rho: G \to GL(V)$ where $V$ is a ...
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### Modulo calculation on a polynomial, in NASA tutorial on Reed-Solomon codes

I am reading Geisel's tutorial$^{\color{red}{\star}}$ on Reed-Solomon codes, in which a Galois Field is developed. The elements of the field are generated as consecutive powers of $X$, modulo an ...
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### Algorithm that solves a system of linear equations over finite fields when a parameter is needed

I was reading Kipnis' and Shamir's paper on Cryptanalysis of the HFE Public Key Cryptosystem by Relinearisation and I wanted to implement the example at the end in Octave without using any additional ...
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### Finding a biggest (in terms of dimension) vector space in a finite set

I previously asked a similar question to this but noticed that the formulation was slightly different than what I am interested in, therefore I ask for any useful information on this problem (by any ...
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### Characteristic and order of a field [closed]

Let $\mathbb{F}$ be a finite field with characteristic $p$, prime. Then $1$ has order $p$ in $(\mathbb{F}, +)$. I don't understand why this then implies that $p$ divides the order of $\mathbb{F}$?
$$\{ a + b i \mid a \in \{0, 1, 2, 3, 4\}, b \in \{0, 1, 2, 3, 4\} \}$$ With calculation done in $\pmod{5}$, I'm wondering if this makes a finite field. I thought the answer is yes at first. Then ...