Exercise 4.4.2

Proof.

(a)

Let $\alpha \in ℂ$ be a root of $$f=x^{11}-(\sqrt{2}+\sqrt{5})x^5+3\sqrt[4]{12}{x}^3+(1+3i)x+\sqrt[5]{17}.$$

Let $L=\mathbb{Q}(\sqrt{2},\sqrt{5},\sqrt[4]{12},i,\sqrt[5]{17},\alpha )$, and $K=\mathbb{Q}(\sqrt{2},\sqrt{5},\sqrt[4]{12},i,\sqrt[5]{17})$.

$f\in K[x]$, and $\alpha$ is a root of $f$. The minimal polynomial $p$ of $\alpha$ over $K$ divides $f$, thus $[L:K]=[K(\alpha ):K]=\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(p)\le \mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(f)=11$:

$$[L:K]\le 11.$$

Moreover, if we write

$K_4=\mathbb{Q}(\sqrt{2},\sqrt{5},\sqrt[4]{12},i),K_3=\mathbb{Q}(\sqrt{2},\sqrt{5},\sqrt[4]{12}),K_2=\mathbb{Q}(\sqrt{2},\sqrt{5}),K_1=\mathbb{Q}(\sqrt{2})$,

then

$$\begin{array}{llll}\hfill [K:\mathbb{Q}]& =[K:K_4].[K_4:K_3].[K_3:K_2].[K_2:K_1].[K_1:\mathbb{Q}]& \hfill & \\ \hfill & =[K_4(\sqrt[5]{17}):K_4].[K_3(i):K_3].[K_2(\sqrt[4]{12}):K_2].[K_1(\sqrt{5}):K_1].[\mathbb{Q}(\sqrt{2}):\mathbb{Q}]& \hfill & \\ \hfill & & \hfill \end{array}$$

The minimal polynomial $P$ of $\sqrt[5]{17}$ over $K_4$ divides $x^5-17\in \mathbb{Q}[x]\subset K_4[x]$, thus $[K_4(\sqrt[5]{17}):K_4]=\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(P)\le 5$. With similar arguments,

$$[K_3(i):K_3]\le 2,[K_2(\sqrt[4]{12}):K_2]\le 4,[K_1(\sqrt{5}):K_1]\le 2,[\mathbb{Q}(\sqrt{2}):\mathbb{Q}]\le 2,$$

Consequently

$$\begin{array}{llll}\hfill [K:\mathbb{Q}]& \le 5\times 2\times 4\times 2\times 2=160& \hfill & \end{array}$$

and

$$[L:\mathbb{Q}]=[L:K][K:\mathbb{Q}]\le 11\times 160=1760.$$

(b)

By Lemma 4.4.2(b), as $\alpha \in L$, the degree of the minimal polynomial of $\alpha$ over $\mathbb{Q}$ divides $[L:\mathbb{Q}]\le 1760$, hence $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(p)\le 1760$.