Exercise 4.3.5

Proof.

(a)

The Schönemann-Eisenstein Criterion with $p=2$ shows that $x^4-2$ is irreducible over $\mathbb{Q}$, and with $p=3$ shows that $f=x^3-3$ is irreducible over $\mathbb{Q}$ (alternatively, we can use Exercise 4.2.8).

(b)

As $x^4-2$ is irreducible over $\mathbb{Q}$ by (a), $x^4-2$ is the minimal polynomial over $\mathbb{Q}$ of $\sqrt[4]{2}$.

$$[L:\mathbb{Q}]=[L:\mathbb{Q}(\sqrt[4]{2})]\times [\mathbb{Q}(\sqrt[4]{2}):\mathbb{Q}],$$

thus $4=\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(x^4-2)=[\mathbb{Q}(\sqrt[4]{2}):\mathbb{Q}]$ divides $[L:\mathbb{Q}]$.

As $x^3-3\in \mathbb{Q}[x]$ is a fortiori in $\mathbb{Q}(\sqrt[4]{2})[x]$, the minimal polynomial $P$ of $\sqrt[3]{3}$ over $\mathbb{Q}(\sqrt[4]{2})$ divides $x^3-3$, so its degree satisfies $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(P)\le 3$. Consequently, $[L:\mathbb{Q}(\sqrt[4]{2})]=\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(P)\le 3$ (et $[\mathbb{Q}(\sqrt[4]{2}):\mathbb{Q}]=4)$, thus

$$[L:\mathbb{Q}]=[L:\mathbb{Q}(\sqrt[4]{2})]\times [\mathbb{Q}(\sqrt[4]{2}):\mathbb{Q}]\le 12$$

(c)

Similarly, $x^3-3$ is the minimal polynomial of $\sqrt[3]{3}$ over $\mathbb{Q}$. $$[L:\mathbb{Q}]=[L:\mathbb{Q}(\sqrt[3]{3})]\times [\mathbb{Q}(\sqrt[3]{3}):\mathbb{Q}],$$

thus $3=\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(x^3-3)=[\mathbb{Q}(\sqrt[3]{3}):\mathbb{Q}]$ divides $[L:\mathbb{Q}]$.

(d)

As $3\mid [L:\mathbb{Q}]$, and as $4\mid [L:\mathbb{Q}]$, where 3 and 4 are relatively prime, $$12=3\times 4\mid [L:\mathbb{Q}].$$

In particular, $12\le [L:\mathbb{Q}]$. By (b), $12\ge [L:\mathbb{Q}]$, thus

$$[L:\mathbb{Q}]=12.$$