Forms of higher degree over certain fields

Let $F$ be a nonformally real field, $n,r$ positive integers. Suppose that for any prime number $p\le n$ the quotient group $F^*/{F^*}^p$ is finite. We prove that if $N$ is big enough, then any system of $r$ forms of degree $n$ in $N$ variables over $F$ has a nonzero solution. Also we show that if in addition $F$ is infinite, then any diagonal form with nonzero coefficients of degree $n$ in $|F^*/{F^*}^n|$ variables is universal, i.e. its set of nonzero values coincides with $F^*$.