Is that an algebraic number?

TheOogy

is [tex]\sqrt{2}+\sqrt{5}[/tex] an algebraic number?
i used 2 and 5 arbitrarily, try any integers (as long as they are not the same integer, in which case it is algebraic)
I tried finding a polynomial with rational coefficients that zeros at this value, but haven't found any.

Hurkyl

If it was, its powers would span a finite dimensional vector space over Q.

matt grime

http://www.dpmms.cam.ac.uk/~wtg10/galois.html

is a useful link to expand on Hurkyl's idea.

Or.

Define x to be that expression above, what is x^2? what is x^2 - 2 - 5?

TheOogy

thanks matt grime!

HallsofIvy

Yes, of course it is. If [itex]x= \sqrt{2}+ \sqrt{5}[/itex] then [itex]x- \sqrt{2}= \sqrt{5}[/itex] and [itex](x- \sqrt{2})^2= x^2- 2\sqrt{2}x+ 2= 5[/itex]. Then [itex]x^2- 3= 2\sqrt{2}x[/itex] so [itex](x^2- 3)^2= x^4- 6x^2+ 9= 8[/itex]. [itex]\sqrt{2}+ \sqrt{5}[/itex] satisfies the polynomial equation [itex]x^4- 6x^2+ 1= 0[/itex] and so is algebraic.

TheOogy

HallsofIvy,
[itex]
(\sqrt{5}+\sqrt{2})^4-6(\sqrt{5}+\sqrt{2})^2+1 = 98.596
[/itex]


i got a different result, for any [itex]\sqrt{a}, \sqrt{b}[/itex]
just use [itex](\sqrt{a}+ \sqrt{b})*(\sqrt{a}- \sqrt{b})*(-\sqrt{a}+ \sqrt{b})*(-\sqrt{a}- \sqrt{b})[/itex] and expand
i haven't read the whole article, just the start and deducted this (without proof) by factoring the polynomial they gave for 2 and 3

HallsofIvy

Thanks for the correction. Here's my mistake:
instead of [itex](x^2- 3)^2= x^4- 6x^2+ 9= 8[/itex] is should have
[itex](x^2- 3)^2= x^4- 6x^2+ 9= 8x^2[/itex]. I dropped the "x" in "[itex]2\sqrt{2}x[/itex]" when I squared.

With that correction, we get [itex]x^4- 14x^2+ 9= 0[/itex] and this time I checked, with a calculator, that [itex]\sqrt{2}+ \sqrt{5}[/itex] satisfies that equation.

Since [itex]\sqrt{2}+ \sqrt{5}[/itex] satisfies [itex]x^4- 14x^2+ 9= 0[/itex], it is algebraic.

JasonRox

If you can prove that the algebraic numbers form a group additively, then you are done.

camilus

jason, what do you mean "form a group additively"? I dont get it, do you mean some sort of commutative property? Although I doubt it..

Moo Of Doom

It means if you add or subtract algebraic numbers from each other, you get an algebraic number.

camilus

Thats a great question. I was working on a similar question, whether e+pi was transcendental.

CRGreathouse

Hah! good luck.

JasonRox

I never said anything about commutativity (even though in this case there is).

JasonRox

Haha, yeah like CRGreathouse said, good luck.

This question is way beyond the calibre of question compared to the one in the OP.

Office_Shredder

The algebraic numbers form a field even, but that's a little tricky to prove (specifically if a and b are algebraic numbers, then a*b is too)