# 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 $\operatorname{GF}(q)$.

5,386 questions
Filter by
Sorted by
Tagged with
70 views

• 67
1 vote
69 views

### Galois group of splitting field of $x^3-5$ over $\mathbb F _7$

Honestly, I'm not even sure where to start. I think I understand how to find the Galois group of a field extension with $\textrm{char}\mathbb F=0$ but for some reason I'm confused when it comes to ...
56 views

### Is every algebraic extension of a finite field Galois?

Let $E/F$ be a (not necessarily finite) algebraic extension, where $F$ is finite. Now, it is known that $E/F$ is a normal extension. On the other hand, $E/F_p$ is algebraic and, since $F_p$ is perfect,...
• 306
57 views

1 vote
35 views

### uniquness of finite fields if they are inbedded in a algebraic closure

I read in the book of Bosch 3.8 after Theorem 2, that if we fix a algebraic closure of $F_p$ all fields of char p with q elements are equal (not just by isomorphism but really equal). His argument is ...
• 311
37 views

### Artin-Schreier extension is cyclic of degree 1 or $p$

Let $K$ be a ﬁeld of characteristic $p > 0$ and $K \subset L$ the extension obtained by adjoining the zeros of the Artin–Schreier polynomial $f = x^p − x − a \in K[x]$, where $a\in K^*$, to $K$. ...
1 vote
43 views

### Example of an Infinite-Dimensional Non-Commutative Division Ring over a Finite Field

According to Wedderburn's little theorem, any finite-dimensional division ring over a finite field must be a commutative division ring, i.e., it is a field. So the question arises: What about infinite-...
• 925
25 views

### What's the point of the local zeta function?

I'm currently reading through Ireland and Rosen's "A Classical Introduction to Modern Number Theory", and I feel I'm missing the point of Chapter 11 on (local) zeta functions. When I see ...
48 views

### Basic concepts about irreducible poly in finite fields

I am a bit confused about the behavior of polynomials in finite fields. Why in a splitting field $\mathbb F2[x]/x^3+x+1$,$x^3=x+1$? I have problem in understanding it intuitively, if α is the root, ...
96 views

• 437
40 views

### Distributing elements of a multi-set to triplets with certain properties possible?

Let $M$ be the multi-set which contains exactly $7$ copies of each positive integer from $1$ to $15$. That is, $M=${$1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,\ldots , 15,15,15,15,15,15,15$}. Is it ...
• 291
101 views

### Ideal of multivariate polynomial

Let $p(x_1,...,x_n)$ be an element of $\mathbb{F}_2[x_1,...,x_n]/(x_1^2+1,...,x_n^2+1)$. Is there a way to capture the size or dimension of the ideal $(p(x))$? I would guess that if there is a way, it ...
• 774
34 views

### Efficient computation of factorial in finite field

What is the state of the art for fast computation of the factorial function $n!$ and more generally of $\prod_{1\leq x\leq n} (x-q)$ (rising/falling factorial) in a finite field? I found just one ...
• 538
49 views

### $A^{p^{n!}}-A$ is nilpotent in $M_n(\mathbb{F}_p)$

As mentioned in the title I want to show $A^{p^{n!}}-A$ is nilpotent matrix for any $n\times n$ matrix $A$ with elements in $\mathbb{F}_{p}$. So far I only know if $\lambda$ is eigenvalue of $A$, then ...
• 913
1 vote