If $a$ is a positive integer that is not a perfect square then $[\Bbb Q(a^{\frac{1}{4}}):\Bbb Q]=4$ If $a$ were a square free number I am done by Eisenstein  criterion but if $a$ is not a perfect square how do I use Eisenstein criterion. The book by Joseph Rotman gives an argument that that there exists a prime $p$ such that $p$ divides $a$ but $p^2$ does not divide $a$, but I dont get this.

May be Rotman means the following.
Consider the prime factorization 
$$
a=\prod_{i=1}^kp_i^{a_i}.
$$
Because 
$$\Bbb{Q}(a^{1/4})=\Bbb{Q}([a/p^4]^{1/4})\tag{*}$$ for any prime $p$, we can without loss of generality assume that $a_i<4$ for all $i$. As we assumed that $a$ is not a perfect square we also know that at least one of exponents, say $a_{i_0}$, is odd. It sounds like you know what to do when $a_{i_0}=1$, so the troublesome case is that of $a_{i_0}=3$.
We can deal with that case by a trick similar to $(*)$. All we need to do is to observe that
$$
\Bbb{Q}(a^{1/4})=\Bbb{Q}(a^{-1/4})=\Bbb{Q}(A^{1/4}),
$$
where
$$
A=\frac{\prod_{i=1}^kp_i^4}a=\prod_{i=1}^kp_i^{4-a_i}.
$$
Here $4-a_{i_0}=1$, so Eisenstein's criterion proves that $x^4-A$ is irreducible over $\Bbb{Q}$ and you are done.
