The ideal-theoretic <existence theorem of global class field theory> gives an inclusion-reversing correspondence between finite abelian extensions $L/K$ and congruence subgroups
$$
P_K(\mathfrak m)\subseteq H\subseteq I_K(\mathfrak m).
$$
The corresponding field satisfies
$$
H=\ker\!\left(I_K(\mathfrak m)
\xrightarrow{\operatorname{Art}_{L/K}}
\operatorname{Gal}(L/K)\right),
\qquad
I_K(\mathfrak m)/H\cong\operatorname{Gal}(L/K).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 2 / d / Solution

The subgroup $H/P_{\mathbb Q}(8\infty)=\{1,7\}$ corresponds inside $\mathbb Q(\zeta_8)$ to the fixed field of $\{1,c\}$, where $c$ is complex conjugation. This is the maximal real subfield
$$
\mathbb Q(\zeta_8+\zeta_8^{-1})
=\mathbb Q(\sqrt2).
$$
Thus the <existence theorem of global class field theory> associates $H$ with $\boxed{\mathbb Q(\sqrt2)}$.

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 2 / e / Solution

Finite abelian extensions of $K$ correspond contravariantly to congruence subgroups $H$ with $P_K(\mathfrak m)\subseteq H\subseteq I_K(\mathfrak m)$, and the Artin map identifies $I_K(\mathfrak m)/H$ with the Galois group.


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