Prove there are no constant polynomials in Zp[x] that have multiplicative inverses.
Favorites|Homepage
Subscriptions | sitemap
HOME > Mathematics > Prove there are no constant polynomials in Zp[x] that have multiplicative inverses.

Prove there are no constant polynomials in Zp[x] that have multiplicative inverses.

[From: ] [author: ] [Date: 11-05-12] [Hit: ]
So p divides an or p divides bm. Say p divides an. The an = 0 in Zp. In this way we continually reduce the degree of the polynomial factors until they are constants.......
Let p be a prime. Are there any nonconstant polynomials in Zp[x], that have multiplicative inverses? Explain your answer.

I know the answer is no, just curious how to get it. Thanks!

-
Let f(x) be a nonconstant polynomial in Zp[x].

If f(x) were invertible, then there exists g(x) in Zp[x] such that f(x) g(x) = 1.

Comparing degrees:
deg (f(x) g(x)) = deg 1
==> deg f + deg g = 0, since p is prime (this is false otherwise).
==> deg f = deg g = 0.
This is a contradiction, because deg f > 0.

I hope this helps!

-
Suppose (an x^n + ... + a0) (bm x^m + ... + b0) = 1 in Zp[x]. Then an * bm = 0 in Zp[x]. So p divides an * bm. So p divides an or p divides bm. Say p divides an. The an = 0 in Zp. In this way we continually reduce the degree of the polynomial factors until they are constants.
1
keywords: multiplicative,polynomials,that,are,constant,no,Zp,Prove,have,inverses,in,there,Prove there are no constant polynomials in Zp[x] that have multiplicative inverses.
New
Hot
© 2008-2010 http://www.science-mathematics.com . Program by zplan cms. Theme by wukong .