For every positive integer b, show that there exists a positive integer n such that the polynomial x^2 ? 1 ? (Z/nZ)[x] has at least b roots.
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August 5, 2017For every positive integer b, show that there exists a positive integer n such that the polynomial x^2 ? 1 ? (Z/nZ)[x] has at least b roots.
1. For every positive integer b, show that there exists a positive integer n such that the polynomial x^2 ? 1 ? (Z/nZ)[x] has at least b roots.
2. Suppose n is a positive integer and there exists a primitive root modulo n. Prove that the only solutions to x^2 ? 1 (mod n) are x ? 1 (mod n) and x ? ?1 (mod n).
