MGF of a multiplication of RV

Hi, I am at a loss for a problem...

If Y1 and Y2 are independent random variables, each having a normal distribution with mean 0 and variance 1, find the moment-generating function of U=Y1Y2. Use this moment-generating function to find E(U) and V(U). Check the result by evaluating E(U) and V(U) directly from the density functions for Y1 and Y2.

I try to do Mu(t) = E(e^tu) the normal way, but then I get: E(e^ty1y2) which doesn't help me in the least...

I know that E(U) = E(Y1)*E(Y2) = 0 and V(U) = 1 similarly, but I cannot find the MGF. Any pointers? Am I missing something obvious, or not so obvious?