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Classical algebra

Groups, Rings, Domains, Modules, etc, Galois theory
Asis ghosh
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Joined: Mon Feb 05, 2018 4:22 am

Classical algebra

solve it

Let $f(x) \in \mathbb{Z}[x]$ be a polynomial of degree $\geq 2$. Pick each correct statement from below:

1. If $f(x)$ is irreducible in $\mathbb{Z}[x]$ then it is irreducible in $\mathbb{Q}[x]$ .

2. If $f(x)$ is irreducible in $\mathbb{Q}[x]$ then it is irreducible in $\mathbb{Z}[x]$.

3. If $f(x)$ is irreducible in $\mathbb{Z}[x]$ then for all primes $p$ the reduction $\overline{f(x)}$ of $f(x)$ modulo $p$ is irreducible in $\mathbb{F}_p[x]$.

4. If $f(x)$ is irreducible in $\mathbb{Z}[x]$ then it is irreducible in $\mathbb{R}[x]$.
Last edited by admin on Tue Feb 20, 2018 5:19 pm, edited 2 times in total.
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