Tagged Questions

5
votes
2answers
104 views

Seeing that $\Bbb F_2[x]/(x^2+x+1)$ is a field

I have some basic question with polynomials appreciate if someone could explain me this. Following is additional and multiplication tables and it is say that this is a field. Have no idea why say ...
6
votes
3answers
77 views

Is there a pattern of the factorization of a polynomial modulo $p$ as $p$ varies

Take $P\in\mathbb{Z}[X]$ and factorize it modulo $p$, where $p$ is a prime. Modulo different $p$'s the factorization varies. Is there a pattern in this variation? I mean, for example, if $P$ is ...
6
votes
1answer
96 views

Finite factor ring

I'm reading the paper "How to use finite fields for problems concerning infinite fields" of Jean-Pierre Serre. In pp. 2, Serre uses the fact that, if $\Lambda\subset\mathbb C$ is a ring finitely ...
8
votes
3answers
146 views

Using Hensel's Lemma to Factor a Polynomial over $\mathbb{Z}_4[x]$

We recently learned about codes over $\mathbb{Z}_4$, and Hensel's Lemma. The lemma is as follows: Let $f(x) \in \mathbb{Z}_4[x]$. Suppose $\mu(f(x)) = h_1(x)h_2(x) \cdots h_k(x)$, where $h_1(x), ...
0
votes
1answer
104 views

Problem related polynomial ring over finite field of intergers

if $f(x)$ is in $F[x]$. $F$ is field of integer mod $p$. $p$ is prime and $f(x)$ is irreducible over $F$ of degree $n$ . prove that $F[x]/(f(x))$ is a field with $p^n$ elements.
5
votes
2answers
115 views

Irreducible polynomial over field of order p

Let $p$ be a prime and $F=\mathbb{Z}/p\mathbb{Z}$ and $f(t)\in F[t]$ be an irreducible polynomial of degree $d$. I need to show that $f(t)$ divides $t^{(p^{n})}-t$ if and only if $d$ divides $n$.
4
votes
2answers
138 views

How to determine if this is a field?

A finite subring $R$ of a field $V$ contains $1$ (so $1$ is an element of $R$). The question is: True or False: The ring $R$ must be a field. I thought that if $R$ was a field it had to be a finite ...
2
votes
1answer
80 views

If $R$ is a ring s.t. $(R,+)$ is finitely generated and $P$ is a maximal ideal then $R/P$ is a finite field

Let $R$ be a commutative unitary ring and suppose that the abelian group $(R,+)$ is finitely generated. Let's also $P$ be a maximal ideal of $R$. Then $R/P$ is a finite field. Well, the ...
4
votes
2answers
126 views

Find all finite fields k whose subfields form a chain: if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$.

Find all finite fields $k$ whose subfields form a chain: that is, if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$. So, I understand that I'm trying to ...
0
votes
2answers
284 views

Quotient Rings of Polynomials Over Finite Fields

I have this question which I don't know how to approach: Let ${F}_{2} = {Z}/2Z$, find representatives for the residue classes of ${F}_{2}[X]$ modulo the polynomial $f(x)$ and compute the ...
2
votes
3answers
251 views

A ring with a subring that is a field

I'm looking for an example of a ring $R$ such that $R$ has no multiplicative identity, but R has a subring $A$ which is a field
5
votes
2answers
486 views

Explicit examples of infinitely many irreducible polynomials in k[x]

My question is the following. Is it possible to give examples of infinitely many irreducible polynomials in a polynomial ring $k[x]$ with $k$ a field? I'm interested in this because I'm ...
2
votes
4answers
1k views

Can you construct a field with 4 elements?

Can you construct a field with 4 elements? can you help me think of any examples?
3
votes
2answers
225 views

For which prime $p$ are the additive groups $\mathbb{F}_p$ and $\mathbb{F}_p^2$ $\mathbb{Z}[i]$-modules?

Homework for my algebra class. Chapter 14, Exercise 7.8 in Artin's Algebra, Second Edition: Let $F = \mathbb{F}_p$. For which prime integers $p$ does the additive group $F^1$ have a structure ...