Please answer exercise 7.14 Definition 7.5.1 Let [?1],, . ,[?elp be all distinct

Question

Please answer exercise 7.14

Definition 7.5.1 Let [?1],, . ,[?elp be all distinct roots of [f(X). in Z/pZ, and let with h(X)lp having no roots in Z/pZ. Then we say that r]p is a root of f(X)lp of multiplicity m. We say that (alp is a simple root im1. Exercise 7.14 Proue that lp is a root of Lf(X)lp of multiplicity m' if and only if Proof of Theorem 7.5.1 The proof is now obvious If f(X)p is non-zero, then its degree is a natural number, and if it has no roots, then we are done (0 S 0). If (X)p does have roots, by the presentation where [h(X)p has no roots we get that miis the number of roots of [f(X) counted with multiplicities. On the other hand, [h(Xp is non-zero as well, and has degree d - (mm.) which should be a natural number as well. So, mm,S d. The theorem is proved. D Exercise 7.15 (1) For any positive integer k, find n and a linear equation unth coefficients in Z/nz which has k solutions in Z/nZ. (2) Suppose f(X) = ao + al X + + adXd has integer coefficients. Prove that f(X)-0 mod p has p solutions if, and only f, (X)p (XP-X) L9(X)p where g(X) is a polynomial with integer coefficients 3) Let d 0 be an integer. Prove that X-1 0 mod p has p-1 solutions if, and only if, p-1d

Explanation / Answer

Definition : A number c is a root of multiplicity k of a polynomial p if we can factor the term (x - c) k out of p but we cannot factor out (x - c)k+1

7.14. all the following things are in Z/pZ

suppose, x' is a root of f(x) of multiplicity m'

then (x-x')m' is a factor of f(x)

i.e. f(x) = (x-x')m' . h(x),

Now, if h(x') = 0 implies, x' is a zero of h(x) , then x' is a zero of f(x) as well.

i.e. x' is a zero of f(x) of multiplicity m'+1, which is a contradiction.

Hence, h(x') is not equal to 0

Now, for the converse part,

Suppose, f(x) = (x-x')m' .h(x) and h(x') is not equal to 0

Which implies, x' is a zero of f(x) of multiplicity at least m'

Now, suppose, x' is a zero of multiplicity m'+1

Then, h(x) is having x' as a zero.

But, h(x') is not equal to 0 implies, x' can not be a zero of h(x) , i.e, x' is a root of multiplicity m' of f(x)

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