For an unramified prime $\mathfrak p$ of $K$ and a prime $\mathfrak P$ of $L$ above it, the Frobenius automorphism is characterized by
$$
\operatorname{Frob}_{\mathfrak P/\mathfrak p}(x)
\equiv x^{N\mathfrak p}\pmod{\mathfrak P}.
$$
In an abelian extension it is independent of $\mathfrak P$. This element is the <Artin symbol>
$$
\left(\frac{L/K}{\mathfrak p}\right).
$$

Solution

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

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{title2=$\left(\frac{L/K}{\mathfrak p}\right)$}
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For an unramified prime $\mathfrak p$ in an abelian extension $L/K$, the Artin symbol is the unique Galois element acting on residue fields by $x\mapsto x^{N\mathfrak p}$.


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 Incoming links: Artin symbol