Minimal polynomial of $\sqrt2 + \sqrt3$ over $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$

Asked Jan 17 '17 at 17:13

Active Jan 17 '17 at 17:13

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I'm new to minimal polynomials and I've tried to find the minimal polynomial of $\sqrt2 + \sqrt3$ over $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$.

I've come up with $x^4-10x^2+1$ over $\mathbb{Q}$ and $x-(\sqrt2+\sqrt3)$ over $\mathbb{R}$ and $\mathbb{C}$.

Can someone tell me if this is correct?

Thank you!

asked Jan 17 '17 at 17:13

Gorid

5

Perfectly correct. – Bernard Jan 17 '17 at 17:18
@Bernard Agree. You could make it answer. – Jeppe Stig Nielsen Jan 17 '17 at 17:19
I don't think it's necessary: the O.P. only asked for checking. – Bernard Jan 17 '17 at 17:20
Checking is also possible here on MSE. – Dietrich Burde Jan 17 '17 at 17:25
In general, for distinct primes $p$, $q$, the minimal polynomial of $\sqrt{p} + \sqrt{q}$ over $\mathbb{Q}$ is $x^4+2x^2(p+q)+(p-q)^2$; $x-(\sqrt{p} +\sqrt{q})$ over $\mathbb{R}$ and $\mathbb{C}$. – RKR Jan 17 '17 at 17:48

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