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You are not logged in. #1 20090828 06:28:15
Number of roots of polynomialsThis can be proved using the equivalent of the division algorithm for polynomials over a field. A simple proof of this using induction on the degree of can be found on p.28 of D.J.H. Garling, A Course in Galois Theory (1986), Cambridge University Press. Note that this applies to polynomials over a field only. This won’t work with rings in general, e.g. and in . However, if is an integral domain, we can apply the division algorithm to polynomials over its field of fractions . Let be a polynomial of degree in . Let be a root of Considering as a polynomial in , we can write where and . Then . Thus . If is another root of distinct from , write where and . Then as . Thus . Continuing this way, we find that if are all the distinct roots of , then we have for some . It follows that cannot have more than roots in since has degree and so cannot be factorized into more than linear factors in . And since a root of polynomial in is also a root of a polynomial in (Gauss’s lemma), our proof is complete. If is not an integral domain, it may be possible for an thdegree polynomial in to have more than roots. For instance, as a polynomial over , the quadratic polynomial actually has 4 roots, namely 1, 3, 5, 7. That’s because is not an integral domain and so we cannot construct its field of fractions. #2 20090829 09:49:08
Re: Number of roots of polynomialsHi Jane; Last edited by bobbym (20090829 10:19:02) In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. 