Show √ 2 + √ 3 algebraic over Q

  • #1
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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##.
 

Answers and Replies

  • #2
35,625
12,160
Looks right.
 

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