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### 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 ...
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### 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 ...
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### $f=X^p-a\in F[X]$ 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 ...
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### $x^p-a$ 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 ...
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### $f(x)=x^p-a$ 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 ...
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### 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 ...