Math Is Fun Forum

Kurre

Member

Registered: 2006-07-18

Posts: 280

Algebraic numbers

Algebraic numbers are numbers that are roots of polynomials with integer coefficients. But what if we let the "polynomial" have rational exponents? Algebraic exponenets? Real exponenets? maybe complex exponents? Will we then get other number sets?
anyone knows?

My first idea was what happaned if we replaced the coeffecients with algebraic numbers, but I found on wikipedia that is has been shown that a root of a algebraic polynomial is again algebraic. Anyone knows how to prove that?

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Algebraic numbers

So, the algebraic number field is algebraically closed, this is a foundamental property of the field, and means exactly that every root of a polynomial with algebraic coefficients is algebraic too. This comes from basic factorisation properties of plynomials and using the fact that every first degree polynomial over the algebraic numbers has an algebraic solution.

---

IPBLE:  Increasing Performance By Lowering Expectations.

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: Algebraic numbers

So, the algebraic number field is algebraically closed, this is a foundamental property of the field, and means exactly that every root of a polynomial with algebraic coefficients is algebraic too. This comes from basic factorisation properties of plynomials and using the fact that every first degree polynomial over the algebraic numbers has an algebraic solution.

oyea, silly me that didnt think of factorization thanks. But still what happens if we let the exponents of the terms in the "polynomial" vary??

ie, an expression of the form

where K is either Q, A, R or C, and S is either Q or A.

How is the set of the roots of such functions related to Q,A,R or C?

Last edited by Kurre (2008-09-05 23:04:31)

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Algebraic numbers

ohh, I understand now, you're talking about the exponents! But then these functions will no longer be polynomilas. I don't have any clue about the structure of such numbers, and it's likely that there's no theory of these at all, because before even thinking about the structure of the set of roots of such all such functions, you need some information about these roots - bounds, number (i suspect there may be infinite number of roots of such a "polynomial" with "finite" degree).
I don't think dealing in general with numbers like

will be easy.
But when we take the exponents to be rational, you may take some common divisor of the rationalexponents and after a suitable substitution, you're again in the good old algebraic field.

---

IPBLE:  Increasing Performance By Lowering Expectations.

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: Algebraic numbers

I don't think dealing in general with numbers like

will be easy.
But when we take the exponents to be rational, you may take some common divisor of the rationalexponents and after a suitable substitution, you're again in the good old algebraic field.

O yes thats true about rational exponents. Altough if we let the exponents be real there must be many more solutions since the set of such functions is not countable, thus the set of the roots cant be countable either.

Actually, considering the following expression:

taking logaritms yields

thus for each

there is a

such that X is a root of

altough we now restricted X to be positive...

Same applies for complex numbers, with 1/r=b+ci,

And its obvious we can choose b and c such that there is a function with X as root for all X.
very well, the problem was rather trivial after all

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Algebraic numbers

the problem was rather trivial after all

Which problem?

---

IPBLE:  Increasing Performance By Lowering Expectations.

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: Algebraic numbers

the problem was rather trivial after all

Which problem?

uhm the question

bhupabhupa

Member

Registered: 2008-10-02

Posts: 1

Re: Algebraic numbers

Hey you all there, I just love Algebra. No matter what the theory of numbers tell you.

===========

bhupa