- #1

mathmari

Gold Member

MHB

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Hey!

I want to show that the polynomial $x^4-2\in \mathbb{Q}[x]$ remains irreducible in the ring $\mathbb{Q}(i)[x]$.

I have done the following:

The polynomial is irreducible in $\mathbb{Q}[x]$ by Eisenstein's criterion with $p=2$.

Then if $a$ is a root of $x^4-2$ then the degree of the extension $\mathbb{Q}(a)$ over the field $\mathbb{Q}$ is $4$ : $[\mathbb{Q}(a):\mathbb{Q}]=4$.

We assume that the polynomial $x^4-2$ is reducible over $\mathbb{Q}(i)$.

Is everything correct so far? If yes, how could we continue to get a contradiction? Do we have to check all possible factorizations of $x^4-2$ ?

(Wondering)

I want to show that the polynomial $x^4-2\in \mathbb{Q}[x]$ remains irreducible in the ring $\mathbb{Q}(i)[x]$.

I have done the following:

The polynomial is irreducible in $\mathbb{Q}[x]$ by Eisenstein's criterion with $p=2$.

Then if $a$ is a root of $x^4-2$ then the degree of the extension $\mathbb{Q}(a)$ over the field $\mathbb{Q}$ is $4$ : $[\mathbb{Q}(a):\mathbb{Q}]=4$.

We assume that the polynomial $x^4-2$ is reducible over $\mathbb{Q}(i)$.

Is everything correct so far? If yes, how could we continue to get a contradiction? Do we have to check all possible factorizations of $x^4-2$ ?

(Wondering)

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