Method of moment generating function normal distribution

Suppose that is normally distributed with mean 0, and unknown variance . Using the method of moment generating functions, show that has a distribution with 1 df.

I know that the mgf of a normal distribution is , so if I was to replace the variable with what was given in the question then I would get:

which looks nothing like the mgf of a distribution.

the other way I was thinking of doing solving this was using the definition, so I would get:

at which point I don't know how to proceed