I understand what it means for a prime number to ramify in a ring of integers of a number field. However, an infinite prime is an archimedean valuation, what does it mean for an archimedean valuation to ramify in a number field?
what does it mean for a prime at infinity to ramify?
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algebraic-number-theory
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3Look at the archimedean valuation that are induced by the different embeddings of the number field into $\mathbb{C}$; each embedding yields an archimedean valuation that "extends" (or "lies over") an archimedean valuation of the base field; the archimedean valuation *splits* if all these extensions are different, and *ramifies* if some of them are the same (just as when you look at the primes $\mathfrak{q}_i$ lying over $\mathfrak{p}$). Consider how to extend the absolute value of $\mathbb{Q}$ to $\mathbb{Q}(i)$, and to $\mathbb{Q}(\sqrt{2})$; the first is an example of ramification. – 2011-02-18
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5Another way of saying it is that a real place (i.e. an embedding of your number field into $\mathbb{C}$ with image in $\mathbb{R}$) ramifies if it extends to a complex place (an embedding into $\mathbb{C}$ with non-real image). A complex place can never ramify in an extension. For an Archimedean place, splitting and being unramified are the same. – 2011-02-18
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5@Keenan I think you should leave that as an answer, simply in the interest of not having unanswered questions. – 2011-02-18