## The Exact Sequence

Because preserves , there is a natural reduction homomorphism

Theorem 9.3.5   The homomorphism is surjective.

Proof. Let be an element such that . Lift to an algebraic integer , and let be the characteristic polynomial of  over . Using Proposition 9.3.4 we see that  reduces to a multiple of the minimal polynomial of (by the Proposition the coefficients of are in , and satisfies ). The roots of are of the form , and the element is also a root of , so it is of the form . We conclude that the generator of is in the image of , which proves the theorem.

Definition 9.3.6 (Inertia Group)   The inertia group associated to is the kernel of .

We have an exact sequence of groups

 (9.2)

The inertia group is a measure of how ramifies in .

Corollary 9.3.7   We have , where is a prime of over .

Proof. The sequence (9.3.1) implies that . Applying Propositions 9.3.3-9.3.4, we have

Dividing both sides by proves the corollary.

We have the following characterization of .

Proposition 9.3.8   Let be a Galois extension with group , and let  be a prime of lying over a prime . Then

for all

Proof. By definition for all , so it suffices to show that if , then there exists such that . If , then , so . Since both are maximal ideals, there exists with , i.e., . Thus .

William Stein 2012-09-24