By Antoine Chambert-Loir

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**Example text**

For every finite extension L of K, let S(L) = HomK (L, Ω), the set of K-morphisms from L to Ω. This is a finite set, of cardinality [L ∶ K], and the group GK acts on it by the formula g ⋅ φ = g ○ φ, for every φ ∈ S(L) and every g ∈ GK ; moreover, the action of GK is transitive. Every morphism of extensions f ∶ L → L′ induces a map f ∗ ∶ S(L′ ) → S(L) which is compatible with the actions of GK . The assignments L ↦ S(L) and f ↦ f ∗ define a contravariant functor from the category of finite extensions of K to the category of finite sets endowed with a transitive action of GK .

An irreducible component of Spec(A) is a closed subset of the form V(p), where p is a minimal prime ideal of A. 5, every prime ideal of A contains a minimal prime ideal of A. 7). — One says that a topological space is noetherian if every strictly decreasing sequence of closed subsets is finite. Equivalently, a topological space is noetherian if and only if every non-empty family of closed subsets has a minimal element. 8). — Indeed, the property for Spec(A) of being noetherian means that every non-empty family of radical ideals of A has a maximal element.

Assume that A is integrally closed in its field of fractions. 42 CHAPTER 1. COMMUTATIVE ALGEBRA Then, for every prime ideal q of B, one has htB (q) = htA (q ∩ A). 5). — Let A be a unique factorization domain and let p be a prime ideal of A. If ht(p) = 1, then there exists a prime element a ∈ A such that p = (a). Proof. — Let a ∈ p be an arbitrary non-zero element. Since p is a prime ideal, a is not a unit, hence it admits a decomposition a = b1 . . b n be a decomposition as a product of irreducible elements.