= Noether normalization by weighted substitutions
{c}
To normalize $k[a_1,\ldots,a_n]$ 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 $k[b_1,\ldots,b_{n-1}]$. Induction proves the <Noether normalization lemma>, including over finite fields.
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