jre1962
Pantsless Prognosticator
- Joined
- Apr 26, 2009
- Posts
- 6,841
Minimal polynomial of sqrt(2)+sqrt(3) over the rationals.
[sqrt(2)+sqrt(3)]^2=5+2sqrt(6)
No second degree minimal polynomial (due to algebraic independence of sqrt(2), sqrt(3) and sqrt(6) over the rationals).
[sqrt(2)+sqrt(3)]^3=11sqrt(2)+9sqrt(3)
No third degree minimal polynomial.
[sqrt(2)+sqrt(3)]^4=49+20sqrt(6)
Therefore, sqrt(2)+sqrt(3) is a root of x^4-10x^2+1.
Alternatively, use [sqrt(2)+sqrt(3)]^2=5+2sqrt(6) to deduce that sqrt(2)+sqrt(3) is a root of (x^2-5)^2-24=x^4-10x^2+25-24=x^4-10x^2+1
Um, if you say so...*has aneurysm*

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