Exercises for Chapter 5

Bordeaux 1 University MHT 933 - master 2

Mathematics 2009-2010

Exercises for Chapter 5

Exercise 1– [Dedekind’s criterion]

We want first to give a proof of the Dedekind’s criterion seen in course. Recall the result.

Theorem 1. Let p be a prime number. Let K =Q(θ), and T ∈Z[X] be the monic, minimal polynomial of θ. Suppose that

T ≡Y

i

P_i^ei (mod pZ[X]),

where the P_i are monic, irreducible and distinct modulo p. Let g =Y

Pi, h=Y

P_i^ei−1, f = (T −gh)/p∈Z[X].

1. Then Z[θ] isp-maximal if and only if gcd(f , g, h) = 1 in F^p[X].

2. Moreover let O⁰ = (I_p : I_p) where I_p is the p-radical of Z[θ]. If U is a monic lift of T /gcd(f , g, h) to Z[X], we have

O⁰ =Z[θ] +1

pU(θ)Z[θ]

and if m = deg gcd(f , g, h) then [O⁰ : Z[θ]] = p^m hence discO⁰ = discT /p^2m.

1. Prove that

pZ[θ] +g(θ)Z[θ]⊂I_p.

2. Using the fact that T is the minimal polynomial of θ over Fp, show that in fact

Ip =pZ[θ] +g(θ)Z[θ].

3. Letx∈ O⁰. Show that there exists A∈Z[X] such that x=A(θ)/p.

4. Show that xp ∈ I_p if and only if g | A and that, if k is a monic lift of g/(f , g) to Z[X], then xg(θ)∈I_p if and only if hk |A.

1

5. Deduce from this part 2 of the theorem and then part 1.

6. With the same notation, let Ri be the remainder of the Euclidean division ofT byP_i. Setd_i = 1 ife_i ≥2 andR_i ∈p²Z[X],d_i = 0 otherwise. Show that in the above theorem we can take U =Q

P_i^ei−di and that Z[θ] is p-maximal if and only if Ri 6∈p²Z[X] for every i such that ei ≥2.

Exercise 2– [pure cubic fields]

Let K = Q(m^1/3) be a pure cubic field, where m is a cubefree integer not equal to ±1. Write m=ab² with a, b squarefree and coprime. Let θ be the cube root of m belonging to K.

1. Show that ifa² 6≡b² mod 9 then ZK admits

1, θ,θ² b

as a Z-basis.

2. Show that ifa² ≡b² mod 9 then ZK admits

1, θ,θ²+ab²θ+b² 3b

as a Z-basis.

3. Letpa prime which does not dividebin the first case and does not divide 3b in the second case. Find the decomposition of X³−m modp and deduce from this the decomposition of pZK as product of prime ideals.

Exercise 3– [quartic fields]

Let m, n be distinct squarefree integers different from 1 and let K be the quartic field Q(√

m,√ n).

1. Compute a Z-basis of ZK.

2. Find the explicit decomposition of prime numbers inK.

Exercise 4– [around Kummer]

We want now to prove the weak version of Kummer’s theorem (which is true even if Z[θ] is not necessarily p-maximal). Recall the statement.

Theorem 2. Let K =Q(θ), T ∈Z[X] the monic minimal polynomial of θ.

If

T ≡Y

i

P_i^ei (mod pZ[X]),

2

where the P_i are monic, irreducible and distinct modulo p. Then pZ^K =Y

i

a_i,

where the a_i =pZK+P_i^ei(θ)ZK are pairwise coprime ideals. Furthermore, if fi is the degree of Pi we have N(ai) = p^eifi and all prime ideals dividing ai

are of residual degree divisible by f_i. 1. Prove that

a⁻¹_i = 1,Y

j6=i

T_j^ej(θ)/p .

2. Following the proof of Kummer’s theorem, establish the above result.

Exercise 5– [units and class group]

Compute the class group, the regulator and a system of fundamental units for the number fields defined by the polynomials

1. P =X²−10, 2. Q=X³+X+ 1, 3. R =X⁴−3X−5.

3