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Is that an algebraic number? |
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May31-09, 06:56 AM
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#1
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TheOogy is
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Is that an algebraic number?
is  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.
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May31-09, 07:04 AM
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#2
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Hurkyl is
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Re: Is that an algebraic number?
If it was, its powers would span a finite dimensional vector space over Q.
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May31-09, 08:22 AM
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Last edited by matt grime; May31-09 at 08:28 AM..
#3
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matt grime is
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Re: Is that an algebraic number?
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?
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May31-09, 12:22 PM
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#4
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TheOogy is
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Re: Is that an algebraic number?
thanks matt grime!
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May31-09, 01:51 PM
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#5
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HallsofIvy is
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Re: Is that an algebraic number?
Yes, of course it is. If  then  and  . Then  so  .  satisfies the polynomial equation  and so is algebraic.
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May31-09, 02:40 PM
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#6
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TheOogy is
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Re: Is that an algebraic number?
HallsofIvy,
i got a different result, for any 
just use  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
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Jun2-09, 02:01 PM
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#7
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HallsofIvy is
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Re: Is that an algebraic number?
Thanks for the correction. Here's my mistake:
instead of  is should have
 . I dropped the "x" in "  " when I squared.
With that correction, we get  and this time I checked, with a calculator, that  satisfies that equation.
Since  satisfies  , it is algebraic.
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Jun9-09, 12:06 AM
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#8
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JasonRox is
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Re: Is that an algebraic number?
If you can prove that the algebraic numbers form a group additively, then you are done.
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Jun9-09, 04:55 PM
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#9
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camilus is
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Re: Is that an algebraic number?
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..
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Jun9-09, 04:59 PM
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#10
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Moo Of Doom is
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Re: Is that an algebraic number?
It means if you add or subtract algebraic numbers from each other, you get an algebraic number.
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Jun9-09, 05:17 PM
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#11
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camilus is
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Re: Is that an algebraic number?
Thats a great question. I was working on a similar question, whether e+pi was transcendental.
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Jun10-09, 12:24 AM
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#12
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CRGreathouse is
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Re: Is that an algebraic number?
Originally Posted by camilus
Thats a great question. I was working on a similar question, whether e+pi was transcendental.
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Hah! good luck.
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Jun10-09, 11:43 AM
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#13
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JasonRox is
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Re: Is that an algebraic number?
Originally Posted by 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..
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I never said anything about commutativity (even though in this case there is).
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Jun10-09, 11:44 AM
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#14
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JasonRox is
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Re: Is that an algebraic number?
Originally Posted by camilus
Thats a great question. I was working on a similar question, whether e+pi was transcendental.
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Haha, yeah like CRGreathouse said, good luck.
This question is way beyond the calibre of question compared to the one in the OP.
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Jun10-09, 02:15 PM
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#15
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Office_Shredder is
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Re: Is that an algebraic number?
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)
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