What is a $\it{nontrivial}\,$ Square Root?
I need to understand the concept behind a non trivial square root. Also how to answer these two questions and how to get to the answer?
Give a non-trivial square root of 30
Give a non-trivial square root in the integers for (mod 143)
Solution 1:
A nontrivial square root of $a^2$ means a root $\,b \ne \pm a,\,$ so the quadratic $\,x^2 - a^2 = (x-a)(x+a)\,$ has $> 2\,$ roots $\,x = \pm a, b.\, $ Then $\,(b-a)(b+a) = 0\,$ but $\,b\pm a \ne 0,\,$ so $\,b\pm a\,$ is a zero divisor.
Generally we can quickly factor $\,n\,$ given any polynomial which has more roots mod $\,n\,$ than its degree, so any nontrivial idempotent or nontrivial square-root will split $\,n,\,$ since it yields a quadratic with $3$ roots. See this answer for a hint on how to find nontrivial square roots mod composite $\,n.\,$ There we find nontrivial idempotents, but the same idea works for square roots.