Show √ 2 + √ 3 algebraic over Q

[HaLAA]

Homework Statement

Show √ 2 + √ 3 algebraic over Q. Find its degree over Q. Prove the answer.

Homework Equations

The Attempt at a Solution

Let $\alpha= \sqrt{2}+\sqrt{3}\in \mathbb{R}$, then $\alpha^4-10\alpha^2+1=0$ which is a root of $f(x)=x^4-10x^2+1$ where $f(x)$ in $\mathbb{Q}[X]$. Apply rational root test, $f(\pm 1)=-8$ which implies $\alpha \notin \mathbb{Q}$.
Also, $f(x)=x^4-10x^2+1=(x^2+2\sqrt{6}-5)(x^2-2\sqrt{6}+5)\notin \mathbb{Q}[X]$, hence $f(x)$ is irreducible in $\mathbb{Q}[X]$ which shows $\alpha$ is degree of $4$.

 

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Looks right.