While number can be checked for primality in O(n^0.5) then why was it considered to be not in P until AKS test? While a basic algorithm to check for primality of a number 'n' [checking if a divides n for all a less than n] would take O(n), AKS test was the breakthrough after which it was placed in P complexity class.
The bit length of $n$ is $\log(n)$, if we forget about the most significant digits, which is always $1$ except for $n=0$. As a function of $t=\log(n)$ you have $n^{1/2}=2^{t/2}$. It is in terms of the bit length of $n$ that AKS runs in polynomial time, while the naive test doesn't.