I am trying to get into algebraic number theory and found the following definition which I would like to understand:
Definition Let $K$ be an algebraic number field. An equivalence class of absolute values on $K$ is called a prime or place of $K$.
Question What is the underlying equivalence relation of absolute values of $K$ in this definition?
I would be grateful for an explanation or a reference that explains that. Thank you!
Two absolute values on a field $K$ are called equivalent if they define the same topology on the field (same as saying they define the same notion of "convergent sequence" in $K$). It can be proved that two absolute values are equivalent if and only if they are positive powers of each other. This does not mean every positive power of an absolute value is in fact an absolute value since a power of one might not satisfy the triangle inequality: if $|\cdot|$ is the usual absolute value on $\mathbf R$ or $\mathbf C$ then $|\cdot|^r$ is an absolute value for $0 < r \leq 1$ but not for $r > 1$ because $|\cdot|^r$ for $r > 1$ does not satisfy the triangle inequality. If $|\cdot|$ is a nonarchimedean absolute value then every positive power of it is a nonarchimedean absolute value.
When $K$ is a number field, it has the following inequivalent absolute values on $K$:
(1) $|\sigma(x)|$ for a real embedding $\sigma \colon K \to \mathbf R$,
(2) $|\sigma(x)|$ for a complex (non-real) embedding $\sigma \colon K \to \mathbf C$, where $\sigma$ and $\overline{\sigma}$ define the same absolute value on $K$ (for technical reasons related to the product formula, we may use $|\sigma(x)|^2$ as "the absolute value" here even though it does not fit the triangle inequality),
(3) the $\mathfrak p$-adic absolute value $|x|_{\mathfrak p} = (1/{\rm N}(\mathfrak p))^{{\rm ord}_{\mathfrak p}(x)}$ for a nonzero prime ideal $\mathfrak p$ in the ring of integers of $K$.
Theorem: Every nontrivial absolute value on $K$ is equivalent to exactly one of the absolute values described above.
Because each nontrivial absolute value on $K$, up to equivalence, corresponds to exactly one of the different (nonzero) prime ideals of the integers of $K$ or a real embedding or a pair of conjugate complex embeddings, an equivalence class of nontrivial absolute values on $K$ is called a "prime" of $K$.