# H3

There exists an absolute constant \(C\ge1\) such that, for every totally real number field \(F\) of degree \(f=[F:\mathbb Q]\), the field
\[
K=F(\mathrm i),\qquad \mathrm i^2=-1,
\]
satisfies
\[
\operatorname{rd}(K)\le2\operatorname{rd}(F),
\qquad
h(K)\le\bigl(C\operatorname{rd}(F)\bigr)^{2f}.
\]

Here \(h(K)\) is the ideal class number of the ring of integers of \(K\). The constant \(C\) is independent of \(F\) and \(f\).

Here \(\operatorname{rd}(L)=|\operatorname{Disc}(L)|^{1/[L:\mathbb Q]}\) is the root discriminant of a number field \(L\), and all logarithms are natural.