Linked Questions

14
votes
3answers
4k views

Irreducibility of a polynomial if it has no root (Capelli) [duplicate]

Let $F$ be a field of arbitrary characteristic, $a\in F$, and $p$ a prime number. Show that $$f(X)=X^p-a$$ is irreducible in $F[X]$ if it has no root in $F$. This answer to a related question ...
2
votes
2answers
430 views

If $X^p-a$ has no zeros in a field $F$ of characteristic $p$ where $a \in F$, is it irreducible? [duplicate]

Let $F$ be a field of characteristic $p>0$ and $a\in F$. I have an easy question which I'm stuck on. If the polynomial $X^p-a$ has no zeros in $F$ then is it irreducible over $F$? Thank ...
0
votes
0answers
734 views

$f=X^p-a\in F[X]$ is irreducible iff $f$ has no root in $F$ [duplicate]

Let $F$ be a field, $a$ an element of $F$ and $p$ prime. How do I prove that $f=X^p-a\in F[X]$ is irreducible iff $f$ has no root in $F$? Honestly, I have no idea how to approach this. Maybe ...
1
vote
1answer
297 views

$x^p-a$ either has a root or is irreducible [duplicate]

My book (A Book of Abstract Algebra, Pinter) is asking me to explain why if $x^p-a$ factors in $F[x]$ then $x^p-a=p(x)f(x)$ where $\text{deg } p,f \le 2$, here $F$ is a field and $p$ is prime. It ...
1
vote
0answers
132 views

$f(x)=x^p-a$ is either ireducible or has a root? [duplicate]

Let $p$ be a prime number. Prove that for any field $k$ and any $a\in k$, the polynomial $f(x)=x^p-a$ is either irreducible or has a root. I think if $\operatorname{Char}k=0$ then $f$ is an ...
1
vote
1answer
64 views

Let $a \in F - F^{p}$. Show that $x^{p} - a$ is irreducible over $F$. [duplicate]

Let $F$ be a field of characteristic $p>0$, and let $a \in F - F^{p}$. Show that $x^{p} - a$ is irreducible over $F$. I didn't get a good idea to solve that question. But I'm not looking for an ...
0
votes
0answers
52 views

prove that $[F(\alpha):F]=p$ where $\alpha$ is a root of $x^p-a$ [duplicate]

This is the problem 1.1 from the book, A Gentle Course in Local Class Field Theory. Let $p$ be a prime, let $F$ be a field of characteristic $\neq p$, and let $a \in F^{\times} \backslash F^{\times p}...
0
votes
1answer
48 views

Show that $x^p - m$ is irreducible for prime $p$ and $m \in \mathbb{Q^{\times}}\setminus \left(\mathbb{Q^{\times}}\right)^p$ [duplicate]

I'm stuck if $m$ is not a prime or has a single prime divider (Then using Eisenstein's criterion), e.g, $m=4$ and $p=5$. Any suggestions?
1
vote
0answers
16 views

Irreducibility of $x^p-a \in \Bbb Q[x]$, where $a(>0) \in \Bbb Q $ with $ a^{1/p} \notin \Bbb Q$ [duplicate]

Let $p$ be a prime, and $a$ be a positive rational number such that $a^{1/p}$ is irrational. Then is the polynomial $x^p-a$ always irreducible in $\Bbb Q[x]$? Intuitively it seems obvious, but I don't ...
12
votes
4answers
352 views

Suppose that $x^5$ and $20x+\frac {19}x$ are rational numbers. Then $x$ is also rational

Let $x\neq0$ be a real number such that $x^5$ and $20x+\frac {19}x$ are rational. How can we prove that $x$ is also rational? (This was a question from the RMO 2019 in India.) My attempt: Let $a,b,c,...
7
votes
2answers
176 views

Is $3$ a prime element of $\mathbb{Z[\eta]}$?

How to check whether $3$ is a prime element or not in $\mathbb{Z[\eta]}$, where $\eta$ is a $17$th primitive root of unity. Also in general how can we check an element is prime or not in $\mathbb{Z[\...
2
votes
1answer
853 views

$x^p-a$ irreducible?

Assume $F$ is a subfield of the complex numbers containing the $p$-th roots of unity. If $\alpha$ is a root of $x^p-a$ for some $a\in F$, and $\alpha \not\in F$ then $x^p-a$ is irreducible. To me it ...
10
votes
1answer
283 views

A prime number $p$ is ramified in $\mathbb{Q}(\sqrt[p]{a})$.

Let $p$ be an odd prime number and $a\in \mathbb {Z}$ with $\sqrt[p]{a}\notin \mathbb{Z} $. Prove that $p$ is ramified in the number field $\mathbb{Q}(\sqrt[p]{a})$. My idea is to apply Dedekind's ...
2
votes
4answers
130 views

$\gcd(a,b)=1, x^a = y^b\Rightarrow x = n^b$, $ y = n^a$ for an integer $n$.

If $ a$, $ b$, $ x$, $ y$ are integers greater than $1$ such that $ a$ and $ b$ have no common factor except $1$ and $ x^a = y^b$ show that $ x = n^b$, $ y = n^a$ for some integer $ n$ greater than $1$...
0
votes
2answers
294 views

If $x^p - a$ is reducible in a characteristic $p$ field, then it has a root.

Let $F$ be a field of characteristic $p$, and consider the polynomial $f(x) = x^p -a$ in $F[x]$. I want to show that $f(x)$ is either irreducible or splits over $F$. It's easy to show that if there is ...

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