For each integer $n \geq 2$ , find a polynomial of degree $n$ with non-rational roots, whose Galois group over $\mathbb{Q}$ is $\mathbb{Z}/2\mathbb{Z}$.
Anybody can help me?
For each integer $n \geq 2$ , find a polynomial of degree $n$ with non-rational roots, whose Galois group over $\mathbb{Q}$ is $\mathbb{Z}/2\mathbb{Z}$.
Anybody can help me?
Just collecting some comments together.
If $n$ is even then
$\prod_{i=1}^{n/2}(x^2-2\cdot2^{2i})$ works otherwise its not possible. For instance when $n=3$ a cubic polynomial with no rational roots is irreducible, so its Galois group is of order at least $3$.