Fix a field F and a polynomial P(x) over F. Do P(x^n) keep factoring further and further as n increases? Sometimes, it is easy to see that they do keep factoring: if, for example, P(x)=x-1, or the field F is algebraically closed. The field F has "good heredity" if any polynomial P(x) over F either has 0 or some roots of unity among its zeros; or has a bound on the number of irreducible factors of P(x^n) independent of n. What fields have good heredity, and why do we care? Come find out!