Is there a systematic method of knowing if $n = \sqrt a $ is or is not a rational number?

Question

Consider $$n = \sqrt a $$ where $a$ is any integer. Is there a rigorous, systematic method of figuring out if $n$ will or will not be a rational number?

Answer 1

The square root of an integer is rational exactly if it is itself an integer.

Namely, suppose $p/q=\sqrt a$. Then $p^2/q^2=a$ which we assume to be an integer. But this means that $q^2$ divides $p^2$.

Now consider the prime factorizations of $p^2$ and $q^2$. Every prime that appears in $q^2$ must appear in $p^2$ with at least the same exponent. But that means that every prime that appears in the prime factorization of $q$ must appear in $p$ with at least the same exponent (each exponent in the prime factorization of $p^2$ is simply twice the corresponding exponent in the factorization of $p$). In other words, $q$ divides $p$, so $p/q$ is an integer.