1
vote
2answers
70 views

When does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$?

Let $p$ be a prime integer, and let $q=p^r$ and $q'=p^k$. For which values of $r$ and $k$ does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$? From Artin's Algebra, Chapter 15, problem 7.12 from the ...
3
votes
2answers
75 views

Finite field with algebraic element over $\mathbb{Z}_p$

Let $F$ be a finite field of characteristic $p$. Show that every element of $F$ is algebraic over $\mathbb{Z}_p$. Since char$(F)$ = $p$, $\forall a \in F $ not zero, $ap$ = 0. We also have the same ...
1
vote
1answer
41 views

Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
0
votes
1answer
38 views

Show that all irreducible polynomials divide $t^{p^d-1} -1$

I have shown that if $f$ is irreducible with $deg(f) = d$ then $\mathbb{F}_p [t] / \langle f\rangle$ is a field with $p^d$ elements. I've also shown that $\langle f\rangle$ is precisely those ...
1
vote
0answers
35 views

kernel of maps associated to the root of an irreducible polynomial

Let $m(\mu)$ be an irreducible polynomial of degree $d$ over $\mathbb{F}_2$, $F_{2^d} = \mathbb{F}_2[x]/(m(\mu))$ by a field extension given by that polynomial and let $d: \mathbb{F}_2[x] \to ...
0
votes
1answer
53 views

Show that a given ring is a field with four elements

Let $R = ( \mathbb{Z} / 2 \mathbb{Z} ) [t]$ be the ring of polynomials with coefficients $\mathbb{Z} / 2 \mathbb{Z}$, $f = f(t) = t^2 + t +1$, and $g = t^2 +1$. Show that: (1) $R/(f)$ is a field with ...
2
votes
2answers
184 views

Find the number of irreducible factors of $x^{63} - 1$

I have to find the number of irreducible factors of $x^{63} - 1$ over $\mathbb{F}_2$ using the $2$-cyclotomic cosets modulo $63$. Is there a way to see how many the cyclotomic cosets are and what is ...
1
vote
2answers
97 views

The proof that a finite field has a prime power order

I don't seem to grasp the proof. First we construct a vector space over a subfield with prime order $p$ where $p$ is the characteristic of the field . As the field is finite , the vector space will be ...
1
vote
2answers
51 views

How to construct finite fields of any prime power order?

For a prime $p$, I know that $\mathbb Z_p$ is a field. To construct a field with four elements, I know I can just take $\frac{\mathbb Z_2[x]}{(x^2+x+1)}$. Similarly, to construct a field of order ...
0
votes
2answers
44 views

Quotient group element is a unit

I'm studying up for my algebra exam, and I'm not exactly sure how to solve a problem like the following Let $f = X^2 + 1 \in \mathbb{F}_5[X]$, $R = \mathbb{F}_5[X]/\langle f \rangle$ and $\alpha = ...
2
votes
1answer
53 views

Find a finite extension of $\mathbb F_2(x,y)$ having infinitely many intermediate subfields.

Let $\mathbb F_2$ be a field of order $2$. Let $F$ be the function field $\mathbb F_2(x,y)$, i.e., $x,y$ are independent variables and $F$ is the field of quotients of the polynomial rings $\mathbb ...
2
votes
0answers
44 views

What's the connection between irreducible polynomials and fixed-frobenius elements in a finite ring?

Consider the ring $A_{p,n} = \mathbb{F}_p [x]/ (x^{p^n}-x)$. It has a basis $\{1, x, x^2, \ldots, x^{p^n - 1}\}$. The Frobenius endomorphism $x \mapsto x^p$ permutes elements of this basis. I've ...
3
votes
1answer
60 views

Units in finite polynomial rings

Are the units of the quotient ring $\mathbb{F}_2[x]/\langle x^k+1 \rangle$ known in general, where $\mathbb{F}_2$ is the finite field with two elements? I'm specifically interested in the case where ...
1
vote
1answer
27 views

Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$.

Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$. It suffices to prove for the case $F=\mathbb{Z}_p$. How to prove?
1
vote
1answer
68 views

finding fixed points of frobenius endomorphism of a ring of char > $p$ (not a domain)

How can I find all the fixed points of the endomporphism $f \mapsto f^p$ in the ring $ \mathbb{F}_p/ (x^{p^n} - x)$. My hunch is that it should have something to do with $x^{p^n}-x$ being the ...
2
votes
3answers
174 views

If $f(x)\in\Bbb Z_p [x]$ is irreducible with $p$ prime and $\deg ⁡f(x)=n$ then $\Bbb Z_p [x]/〈f(x)〉$ is a field with $p^n$ elements.

Suppose that $f(x)\in\Bbb Z_p [x]$ is irreducible, where $p$ is a prime. If $\deg ⁡f(x)=n$,show that $\Bbb Z_p [x]/〈f(x)〉$ is a field with $p^n$ elements. I've seen plenty of cases, looking at a ...
12
votes
0answers
108 views

Ring structure on the Galois group of a finite field

Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure ...
1
vote
3answers
116 views

polynomial ring over finite field

Can someone provide a proof for the following? I wrote a one page proof. I must be doing something wrong. There has to be a quicker way to prove this. Thank you ahead of time. I will highly rate for ...
1
vote
2answers
38 views

$\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,where $c \in \mathbb Z_9$ and $w=e^{2\pi i/9}$

$$\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$$ where $c \in \mathbb Z_9$, $w=e^{2\pi i/9}$ and $\mathbb Z_9$ is the ring of integers modulo 9.
0
votes
1answer
67 views

Example of Linear System of Polynomials over finite field

I'm trying to find any system of polynomial over finite field (solvable in $K[x]_{m(x)}$) with characteristic two. I want please any example of $a_{ij}$ and $b_k$ ...
1
vote
1answer
60 views

Finding a root and reduce a function over a finite field

Let $F= \Bbb Z_2[x]/ \langle f \rangle$ with $f=x^3+x+1 \in \Bbb Z_2[x]$. Now consider f as an element of $F[x]$ and a) show that there exists $\alpha \in F$ with $f(a)=0$ b) find $g \in F[x]$ with ...
1
vote
0answers
142 views

Understanding a proof of Wedderburn's little theorem

I am working on the proof of Wedderburn theorem and I have a problem to understand the part of it. I don't understand why $b_{1}^{-1}a_{1}=\lambda^{i}$ implies the last contradiction ...
0
votes
1answer
84 views

Wedderburn's theorem

I am trying to understand the proof of Wedderburn theorem and I have a problem to understand the part of it. I don't know how to get the equation nr 1: $a_{1}b_{1}=\mu b_{1}a_{1}$ Could anyone ...
5
votes
2answers
122 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
156 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 ...
7
votes
1answer
153 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
286 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
148 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.
6
votes
2answers
172 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
181 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
126 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 ...
9
votes
3answers
220 views

When is $\mathbb{F}_p[x]/(x^2-2)\simeq\mathbb{F}_p[x]/(x^2-3)$ for small primes?

I've been considering the rings $R_1=\mathbb{F}_p[x]/(x^2-2)$ and $R_2=\mathbb{F}_p[x]/(x^2-3)$, where $\mathbb{F}_p=\mathbb{Z}/(p)$. I'm trying to figure out if they're isomorphic (as rings I ...
4
votes
2answers
144 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
3answers
526 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
375 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
8
votes
4answers
781 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 ...
4
votes
4answers
2k 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?
5
votes
2answers
243 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 ...