COMPUTATION OF HILBERT SEQUENCE FOR COMPOSITE QUADRATIC EXTENSIONS USING DIFFERENT TYPE OF PRIMES IN Q

First, we will give all necessary definitions and theorems. Then the definition of a Hilbert sequence by using a Galois group is introduced. Then by using the Hilbert sequence, we will build tower fields for extension K/k, where K=k(d1,d2) and k=Q for different primes in Q.


INTRODUCTION
Let K/k be an extension of degree n.We consider the tower of fields and a tower of integer rings for this extension K_..._L...Dk_ OK _ _ OL _ Ok (]" ]) A prime ideal P in K determines a prime PLin each field of the tower, where each PL is divisible by P. Let p be a rational prime that is divisible by all these prime ideals PL.Then we have: PL Pk n OL p= PL U Z.
If the prime ideal p in k does not split imo n distinct factors of P in K, how far can we go in terms of an intermediate field where splitting occurs?This will be answered later.
First we define what is meant by order and degree DEFINITION 1.1.
(a) Order Pip e Pelp, pe+l Xp (b) Degree P/p f Nk/kP PI LEMMA 1.2.Both order and degree are multiplicative Order P/p order P/ PL order Pz,/p Degree P/p degree PPL" degree PL/P Let us assume here that K/k for [K; k] n is a normal extension.This makes K/L normal for each L in the tower but not in Lilt.Let p have factors P) in L for j 1, 2, 3, g, p =, i=1 n e.f.g.
(1 2b) Let order K/k P e and degree K/k P f.Then for P p, we have order p degree p 1 from ktok.
Thus from k to K the order has grown from to e and the degree has grown from to f and the number of factors in (1.2a) and (1.2b) has grown from to g.We arrange the tower fields in 1.1 in such a way that will separate the growths for K/k normal.
Let K z be a maximal L in {L K _ L _ k}.K z is called the "splitting" field of P in K/L and is such that.
degree PL/P 1 order Pr./p 1 Let us assume that Kr is a maximal L in {L" K _ L _ k}.Kr is called the "inertial" field of P in K/z and is such that degree PL/P fL > 1 order PL/p= 1.
This maximality process can be performed again for all L such that: degree PL/P fL > 1 order PL/p=eL for (eL,p)= l.
The maximal field here is called the "first ramification" field K,, For this field, FL f and eL is a part of e prime to p.This part is called "tame ramification If order e is divisible by p, the ramification is called "wild."Thus we have the new tower fields for extension K/k: It is easier to define 1.2c by the Galois group methods.
DEFINITION 1.2.Let K/k be a normal extension.The Hilbert sequence for an ideal P in K is given by the subgroups of G Gal(K/k) as follows kT {u G" P" =-A mod p} (Go)   (1.3c) kv,.c {u G A =-Amodp "+I}=G,, (r>_O). (1.3d) Where A is an arbitrary integer in Ok.Since G z fixes P, then GT, Gv, and so on are invariant subgroups of Gz.Since Gz preserves P, it is one of g conjugates, IG/Gzl , (1.3e) also, since GT preserves each residue class mod P, IGZ/GTI I(Og/p)/(Ok/)l Ic(f)l-f, which refer to the cyclic Galois group of an extension of a finite field.Furthermore IGTI e.
d. p P for p odd, p[d, pld, plds, (d3/p) 1 with the same proof as above, the following tower fields are produced.
K z Q, KT k3, and K K K K, K.
g. p for dl S (mod 16), d3 1 (mod 8) has the same tower fields as e.
We showed in the above cases, if the prime ideM p of k does not split into n distinct prime factors of K, how we cm build imeediate fields Kz, KT, K, where splicing of prime p occurs.