Mathematics desk
< September 23	<< Aug \| September \| Oct >>	Current desk >

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

September 24

Injective and surjective polynomials

Let K be an algebraically closed field. Does a non-constant polynomial with coefficients in K necessarily define a surjective function from K to itself? Does a polynomial with coefficients in K defining an injective function from K to itself necessarily have degree 1? GeoffreyT2000 (talk, contribs) 22:58, 24 September 2016 (UTC)[reply]

Yes to both questions: (1) if f(x) is a polynomial and a is in K, then f(x)-a splits. (2) A degree n>1 polynomial that is one-to-one would have the property that f(x)-a is a perfect nth power for all a in K. In particular, f(x)=x^n, and x^n-1=(x-1)^n. This implies that n is a power of the characteristic of K, so f (x) is either the identity or is the Frobenius automorphism of a finite field. No finite field is algebraically closed, a contradiction. Sławomir Biały (talk) 23:44, 24 September 2016 (UTC)[reply]

f(x) can also be the Frobenius automorphism of an infinite field of positive characteristic. And you can multiply it by a nonzero constant, and add a constant to it --77.125.79.22 (talk) 21:22, 30 September 2016 (UTC)[reply]