Recently Professor Daqing Wan studied the algebraic degrees of the exponential sums $S_q(f)$ over a finite field $\mathbb{F}_q$. In this article, basing on Wan's results, we discuss the Gauss sums for the case $q=p^2$ and $p\equiv 1\pmod 4$ and obtain that $S_q(x^d)$ has only two possible values, if it is of degree 1. We generalise the method proposed by Myerson and get the explicit values of the algebraic degrees of Gauss sums in some special cases.
