All Questions
53
questions
0
votes
1
answer
27
views
Determining the Equality of Two Field Extensions
Let $F$ be a field of characteristic $0$. Let $F(\alpha)/F$ be a finite extension of degree not divisible by $3$. Is is true that $F(\alpha^3)=F(\alpha)$? If we assume that they are not equal, since $\...
- 730
2
votes
1
answer
127
views
Irreducibility of a Polynomial with Prime Exponents
Let $f(x) = (x^p - a_1)(x^p - a_2) \ldots (x^p - a_{2n}) - 1$
where $a_i \geq 1$ are distinct positive integers where at least two of them are even, and $n \geq 1$ is a positive integer and $p$ is ...
- 637
0
votes
0
answers
69
views
Reducibility of $x^2-7$ over $\mathbb{Q}(\sqrt[5]{3})$
Suppose for a contradiction that $x^2-7$ is reducible over $\mathbb{Q}(\sqrt[5]{3})$. Then $\sqrt{7}\in\mathbb{Q}(\sqrt[5]{3})$. It follows that $\mathbb{Q}\subset\mathbb{Q}(\sqrt{7})\subset\mathbb{Q}(...
- 2,066
0
votes
0
answers
123
views
Show: $f=x^p - \alpha$ irreducible and inseparable in field $F(\alpha)$, where $charF=p>0$ and $\alpha$ transcendental over F
To show it is irreducible I just show that $\sqrt[p]\alpha \notin F$, since it is a zero of the polynomial in question and, if it is not, then it cannot be reducible. (Correct me, if I am wrong here. ...
- 27
1
vote
1
answer
102
views
Is every polynomial of even degree reducible after some field extensions of degree 2?
Given an irreducible polynomial $p \in \mathbb{Q}[x]$ we're interested in how it factors after repeated simple field extension of degree $2$. So we generate a chain of fields $F_{n+1} / F_{n}$, where $...
- 11
-2
votes
2
answers
145
views
Factoring polynomials modulo 3 [closed]
Let $f(x) = x^5 + 2x^2 + 2x + 2 \in\mathbb Z_3[x]$. Then the irreducible factorization of $f(x)$ is $(x^2 +1)(x^3+2x+2)$ even though it does not have a root in $\mathbb Z_3$. How did we find that ...
0
votes
1
answer
405
views
Irreducible polynomial in integers modulo p
I am a completing a past paper question and I am undecided on what method to use here. The question is:
For what $a$ is $f(x)=x^3+x+a\in\mathbb{Z}_{7}[x]$ irreducible? My ideas are:
(1) Check each $a\...
- 99
2
votes
1
answer
201
views
Splitting field of a polynomial over $\mathbb{Z}_p$
For $p$ prime, how can we show that the splitting field of $f(x)=x^{p-1}-1$ over $\mathbb{Z}_p$ coincides with $\mathbb{Z}_p$?
The roots of $f(x)$ are the $p-1$ roots of unity. I think I am correct in ...
- 99
2
votes
0
answers
54
views
Isomorphism of Galois Groups
$E$ is the splitting field for $f(x)=x^3-2$ and $K := E(\sqrt 5)$. We want to show that $G=\text{Gal}(K/\mathbb{Q}) \cong \mathbb{Z_2} \times S_3$.
To do this, we know that $K$ has subfields $E$ and $...
- 99
2
votes
2
answers
314
views
Maximal degree of irreducible polynomials
This is a question I have thought about for a while. We know that every polynomial $p \in \mathbb C[z]$ can be written as a product of monomials
$$p(z) = a \displaystyle\prod_{i=1}^n(z-z_i).$$
Now for ...
- 1,397
2
votes
1
answer
136
views
Invertible elements of $K[x]/\langle p(x)\rangle$ when $p(x)$ is irreducible
I want to find the inverse of $f(x)=ax^{2}+bx+c$ in $L=K[x]/\langle p(x)\rangle$ when $p(x)$ is an irreducible polynomial in $K[x]$ with degree $3$.
I know elements of $L$ are like $f(x)$ such that $...
- 57
4
votes
1
answer
133
views
Minimal polynomial of $x=\sqrt{2}+i\sqrt{3}$
I was asked to calculate the minimal polynomial of $x=\sqrt{2}+i\sqrt{3}$ over the fields
\begin{align*}
K_1 = \mathbb{Q}, \quad K_2 = \mathbb{Q}(\sqrt{2}), \quad K_3 = \mathbb{Q}(i\sqrt{3}), \...
- 1,604
2
votes
1
answer
90
views
A problem about irreducible polynomials over a finite field.
Problem
Let $F$ be a finite field with $q$ elements. Let $f\in F[x]$ be an irreducible polynomial. Prove that if $f \mid x^{q^n}-x$ then $\deg{f}\mid n$ (the converse is also true and I have a proof).
...
- 1,336
0
votes
1
answer
172
views
Irreducible cubic polynomial in $\mathbb{Q}[x]$ has no roots in $\mathbb{Q}(\sqrt{2}, 5^{1/4})$
I am working on the following problem:
Consider the field extension $F = \mathbb{Q}(\sqrt{2}, 5^{1/4})$ of $\mathbb{Q}$. Let $f(x)$ be an irreducible cubic polynomial in $\mathbb{Q}[x]$. Prove that $...
- 1,503
2
votes
1
answer
190
views
Splitting field of $f(x)=x^7-3$ over $\mathbb{Q}$ and its degree
The roots of $f$ has the form $\sqrt[7]{3}\,\gamma^{n}$, where $\gamma=\cos\frac{2\pi}{7}+i\sin\frac{2\pi}{7}$ and $n\in\{0,1,2,3,4,5,6\}$.
I saw that, if $K$ is the root field of $f$ over $\mathbb{Q}...
- 2,666