 Noether normalization by weighted substitutions

To normalize $k[a_1,\dots ,a_n]$ over an arbitrary field, choose a nonzero polynomial relation and a base $M$ larger than all its exponents. The substitutions $b_i=a_i-a_n^{M^i}$ give distinct top weights to its monomials, so its highest power of $a_n$ has a nonzero scalar coefficient. It becomes a monic polynomial equation, making the algebra finite over $k[b_1,\dots ,b_{n-1}]$. Induction proves the Noether normalization lemma, including over finite fields.