## The Exact Sequence

There is a natural reduction homomorphism

Theorem 14.1.5   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 14.1.4 we see that  reduces to the minimal polynomial of (by the Proposition the coefficients of are in , and satisfies , and the degree of equals the degree of the minimal polynomial of ). 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 14.1.6 (Inertia Group)   The is the kernel of .

Combining everything so far, we find an exact sequence of groups

 (14.1)

The inertia group is a measure of how ramifies in .

Corollary 14.1.7   We have , where is a prime of over .

Proof. The sequence (14.1.1) implies that . Applying Propositions 14.1.3-14.1.4, we have

Dividing both sides by proves the corollary.

We have the following characterization of .

Proposition 14.1.8   Let be a Galois extension with group , let be a prime 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 , we have , so since both are maximal ideals, there exists with , i.e., . Thus .

Figure 14.1.2 is a picture of the splitting behavior of a prime .

William Stein 2004-05-06