Congruence with Prime Modulus
Arithmetic in Zp
We saw that a linear congruence ax≡b(modn) has a unique solution (modn) if gcd(a,n)=1. Now if n is a prime p, then gcd(a,n)=gcd(a,p) is either 1 or p; in the first case, we have a unique solution (modp),while in the second case (where p|a), either every x is a solution (when p|b) or no x is a solution (when p∤b).
One can view this elementary result as saying that if the polynomial ax−b has degree d=1 over Z, (that is, if a≠0(modp)), then it has at most one root in Zp. Now in algebra we learn that a non-trivial polynomial of degree d, with real or complex coefficients, has at most d distinct roots in R or C; it is reasonable to ask whether this is also true for the number system Zp, since we have just seen that it is true when d=1. Our first main theorem, due to Lagrange, states that this is indeed the case.
Theorem:Let p be prime and let f(x)=adxd+…+a1x+a0 be a polynomial with integer coefficients, where ai≢0(modp) for some i. Then the congruence f(x)≡0(modp) is satisfied by at most d congruence classes [x]∈Zp.
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