This post is based on Marcus’s Number Fields. More specifically, it is based on a series of exercises following chapter 4.
Recall the definition of the intertia group of a prime in lying over a prime in ( is a Galois extension of number fields) – it’s the set of all such that, for all , we have . We now generalize this group.
Definition: In setting as above, we define the th ramification group to be the set of all such that . The groups are called the higher ramification groups.
It is straightforward to see that , all the subgroups are normal in and their intersection is trivial. The structure of groups can be somewhat complicated, but the groups are particularly simple:
Proposition 1: is isomorphic to a subgroup of .
Proof: Fix . We can then factor as with relatively prime. Taking any we can find, by Chinese remainder theorem, a solution to . Because , so , so for some . In particular, . Also, is well-defined modulo : If , then , so .
Thus we have defined a mapping , and clearly , in particular – this map is a homomorphism into . To show that this it induces the desired isomorphism we need to show that its kernel is , which will easily follow if we show that if , then , i.e. . We will prove something more general:
Lemma 1: For and , if , then .
Proof of the lemma: We will proceed by induction on . This is immediate for . Suppose now . In particular, , so . Therefore for all , so (for last congruence, recall ). So for all .
Now we show the congruence for . Let (as in the proof of the proposition). Choose . Then , so by above and since . But is a unit modulo , hence modulo , so .
At the same time, every conguence class modulo has an element which is fixed by , and indeed, by every element of . By result from my previous post, is a trivial extension of , so every congruence class modulo has a representative in , and by definition these are fixed by elements of . So every can be written as , so that . Therefore .
Hence, as we said, is the kernel of constructed homomorphism, which therefore is an isomorphism of onto its image, which is a subgroup of .
In a quite similar way we can prove the following result:
Proposition 2: is isomorphic to a subgroup of the additive group .
Proof: Let, as before, . Take any . Writing (again) , choose . Then , so for some . Like in the previous proposition, we easily see that is uniquely defined modulo and . This gives us a homomorphism, and from the lemma we easily find that its homomorphism is , so that we get the desired isomorphism from to a subgroup of .
A quite immediate corollary is the following.
Theorem 1: Groups are solvable.
Proof: We consider the chain of normal subgroups . is isomorphic to the Galois group of the finite field , is isomorphic to a subgroup of the multiplicative group of this field and is isomorphic to a subgroup of its additive group. All of these are abelian, and the chain eventually terminates (eventually are trivial), so all the groups in the chain are solvable.
Definition: Suppose a prime in ramifies in and let and be a prime in lying under . We say that wildly ramifies if , and we say that it tamely rafimites otherwise.
The terminology above might seem unmotivated, but hopefully it is at least in part clarified by the following theorem.
Theorem 2: If a prime is ramified, then it’s tamely ramified iff all the higher ramification groups are trivial. Moreover, is a Sylow -subgroup of .
Proof: Since are isomorphic to subgroups of , which is a -group, their sizes are powers of . Hence is a power of , i.e. is a -group. On the other hand, is indivisible by , so must be the Sylow -subgroup of . In particular, it’s nontrivial iff .
The next result will turn out to be rather useful later.
Proposition 3: Suppose is abelian. The embedding from the proof of proposition 1 actually sends into .
Proof: Suppose and . First we note that this implies, in a way similar to the first two paragraphs of the proof of lemma 1, that for all .
Abelianness of implies that, for any other , , so for all . Taking and noting this gives , therefore . Since maps surjectively onto Galois group of , this group acts trivially on , so . .
Higher ramification groups, especially the last proposition, will turn out to be very useful in a proof of Kronecker-Weber theorem, which will be the subject of an upcoming blog post.