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## 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##.