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