Finite extension stability of C1 fields
= Finite extension stability of C1 fields
For a degree-$m$ extension $L/K$, substitute a $K$-basis of $L$ into a homogeneous degree-$d$ polynomial in $N>d$ variables over $L$, then take its <field norm>. The resulting degree-$md$ polynomial has $mN>md$ variables over $K$. A nonzero zero over a <C1 field> gives a nonzero zero of the original form because the norm of a field element vanishes only at zero. Separability is unnecessary.