Prove that $\sqrt{2+\sqrt 3}$ is irrational

Question

Prove that $\sqrt{2+\sqrt3}$ is irrational. I can't seem to figure this one out.

Suppose it is rational. What expressions does this assumption lead to? — abiessu, Oct 02 '15 at 01:44
"I can't seem to figure this one out." Which other ones can you figure out? — Did, Oct 02 '15 at 06:27

score 5 · Answer 1 · answered Oct 02 '15 at 01:54

5

Hint: $x\in \mathbb{Q} \implies x^2 \in \mathbb{Q}$

answered Oct 02 '15 at 01:54

Reveillark

1

(+1) However, one needs to know that $\sqrt3\not\in\mathbb{Q}$. – robjohn Oct 02 '15 at 03:25

score 2 · Answer 2 · edited Apr 13 '17 at 12:21

2

$\sqrt{2+\sqrt3}$ satisfies $x^4-4x^2+1=0$, which means that $\sqrt{2+\sqrt3}$ is an algebraic integer.

As shown in this answer, if $x$ is an algebraic integer, then $x\in\mathbb{Q}\implies x\in\mathbb{Z}$.

Since $1\lt\sqrt{2+\sqrt3}\lt2$, $\sqrt{2+\sqrt3}\not\in\mathbb{Z}$; therefore, $\sqrt{2+\sqrt3}\not\in\mathbb{Q}$.

edited Apr 13 '17 at 12:21

Community

answered Oct 02 '15 at 03:24

robjohn

2 Answers2