What is the definition of "prime ideal decomposition"?

Question

I'm reading about Sunada's theorem in the book Geometry and Spectra of Compact Riemann Surfaces (Peter Buser) and I encountered this paragraph:

If R is an algebraic number field and if $p \in \mathbb{N}$ is a prime number, then $p$ has a unique prime ideal decomposition with respect to $R$. Let $d_1(p), \ldots, d_v(p)$ be the degrees of the prime ideal factors, arranged such that $d_1(p) \leq \cdots \leq d_v(p)$ and call the finite sequence $$\ell[R](p) = (d_1(p), \ldots d_v(p)) $$ the length of $p$ with respect to $R$. How much information about $R$ can we read out of these lengths?

(bold emphasis mine). The term prime ideal decomposition isn't defined anywhere in the book, and I haven't been able to find a definition online. Can someone give me a definition, or point me towards a reference?

Answer 1

In a Dedekind domain every ideal is (in a unique way) the product of prime ideals. (The product of two ideal is the ideal generated by all products of elements.)

The ring of integers of an algebraic number field is a Dedekind domain. The prime ideal decomposition of $p$ is the factorization of the principal ideal generated by $p$ into prime ideals of this Dedekind domain.

Answer 2

It can be shown that rings of algebraic integers algebraic numbers which are roots of a monic polynomials are so-called Dedekind domains – a generalisation of both PIDs and UFDs. They have several characterisations, most notably that ideals have a decomposition as a product of prime ideals, and this decomposition is unique, up to the order of the factors.

The main difference with PIDs is that non-zero prime ideals need not have a single generator, and that irreducible elements need not generate a prime ideal.