infinite primes

Question

I am reading these notes by milne. http://www.jmilne.org/math/CourseNotes/CFT310.pdf. In it page 4 example 0.5 for instance he talks about "both infinite primes." Does anyone know what this means? Thanks.

Answer 1

Quote from page 2:

By a “prime” of $K$, we mean an equivalence class of nontrivial valuations on $K$. Thus there is exactly one prime for each prime ideal in OK, for each embedding $K \to\mathbb R$, and for each conjugate pair of nonreal embeddings $K \to\mathbb C$. The corresponding primes are called finite, real, and complex respectively. An element of K is said to be positive at the real prime corresponding to an embedding $K \to\mathbb R$ if it maps to a positive element of $\mathbb R$. A real prime of $K$ is said to split in an extension $L/K$ if every prime lying over it is real; otherwise it is said to ramify in $L$.

So the infinite primes, I'm guessing, correspond to the real and complex primes. In this case, there are only real primes, and there are two of them. These correspond to the two embeddings of $\mathbb Z[\sqrt 6]$ into $\mathbb R$, with the normal valuation on $\mathbb R$.