Abstract Algebra II · Finite Galois theory
A quartic whose splitting field is only quadratic
Problem
Find the splitting field over $\mathbb Q$ of $f(x)=x^4+4$, compute its degree, and identify its Galois group including the action on all four roots.
Hint
Use $x^4+4=(x^2-2x+2)(x^2+2x+2)$.
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