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Show √ 2 + √ 3 algebraic over Q

  1. Jun 3, 2015 #1
    1. The problem statement, all variables and given/known data
    Show √ 2 + √ 3 algebraic over Q. Find its degree over Q. Prove the answer.

    2. Relevant equations


    3. 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##.
     
  2. jcsd
  3. Jun 4, 2015 #2

    mfb

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    2016 Award

    Staff: Mentor

    Looks right.
     
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