Find the automorphisms of the extension $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}$ field explicitly.
It is well knowned that the field $\mathbb{Q}(\sqrt[4]{2})$ is not Galois over $\mathbb{Q}$ since any automorphism is determined by where it sends $\sqrt[4]{2}$ and of the four possibilities $\{±\sqrt[4]{2},±i\sqrt[4]{2}\}$, only two are elements of the field. And since $\mathbb{Q}(√2)/\mathbb{Q}$ and $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}(√2)$ are quadratic extension so galois extension. As we know that any extension $K$ of $F$ of degree $2$ is of the form $F(√D)$ where $D$ is the element of $F$ which is not a squre in $F$. Then how can we express $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}(√2)$ as like $F(√D)$ and what are the automorphisms of the extension $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}$. Please help me to solve this.
There are two automorphisms, the identity, and the automorphism mapping $2^{1/4}$ to $-2^{1/4}$. It's not a Galois extension, so doesn't have four automorphisms. The only conjugates of $2^{1/4}$ in $\Bbb Q(2^{1/4})$ are $\pm 2^{1/4}$.