Cubic Polynomial $x^3-2$ is Irreducible Over the Field $\Q(i)$
Problem 399
Prove that the cubic polynomial $x^3-2$ is irreducible over the field $\Q(i)$.

Prove that the cubic polynomial $x^3-2$ is irreducible over the field $\Q(i)$.
Let $\Q$ be the field of rational numbers.
(a) Is the polynomial $f(x)=x^2-2$ separable over $\Q$?
(b) Find the Galois group of $f(x)$ over $\Q$.
Prove that the quadratic fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are not isomorphic.
Determine the splitting field and its degree over $\Q$ of the polynomial
\[x^4+x^2+1.\]
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Show that the polynomial $x^3-\sqrt{2}$ is irreducible over the field $\Q(\sqrt{2})$.