Applications of additive version of Hilbert's theorem 90

Solution 1:

An example that first comes to my mind deals with quadratic equations over a finite field $\Bbb{F}_{2^n}$ of characteristic two. There the additive Hilbert 90 says that $$ x^2+x=a $$ with $a\in \Bbb{F}_{2^n}$ has a solution (obviously then two solutions) in $\Bbb{F}_{2^n}$, if and only if $tr(a)=0$. This reinterpretation comes from the following observations:

• Squaring, i.e. the Frobenius automorphism $F$, is a generator of the Galois group $Gal(\Bbb{F}_{2^n}/\Bbb{F}_2)$.
• $F(x)-x=x^2-x=x^2+x$ for all $x\in \Bbb{F}_{2^n}$.

This leads to solvability criteria of a general quadratic over $\Bbb{F}_{2^n}$: $$ x^2+bx+a=0\qquad(*) $$ with $a,b\in\Bbb{F}_{2^n}$ has solutions in $\Bbb{F}_{2^n}$, iff $tr(a/b^2)=0$ - divide $(*)$ by $b^2$, and write it in terms of the new variable $y=x/b$). Note that the usual trick of completing the square is unavailable in characteristic two. Also note that $(*)$ has a double root in $\Bbb{F}_{2^n}$, if $b=0$.

Admittedly this example is not very satisfying here, because:

• we can derive this result without knowing about Hilbert 90, and
• it's not about algebraic number fields.

Solution 2:

Google Artin-Schreier theory, an additive analogue of Kummer theory. In particular, the proof of the Artin-Schreier theorem about non-algebraically closed fields whose algebraic closure is a finite extension is a beautiful application (when treating the case of fields with characteristic $p$).

Quite generally, these additive theorems are useful in characteristic $p$ when dealing with extensions of degree $p$ or more generally a power of $p$, depending on the situation. If you are not a fan of characteristic $p$ then the additive Hilbert Theorem 90 will not be your friend.