_{1}

Theoretical foundations of a new algorithm for determining the p-capitulation type ù(*K*) of a number field K with p-class rank ?=2 are presented. Since ù(*K*) alone is insufficient for identifying the second p-class group G=Gal(F_{p}^{2}*K*∣*K*) of K, complementary techniques are deve- loped for finding the nilpotency class and coclass of . An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(*K*)=(*τ* (*K*),ù(*K*)) of all 34631 real quadratic fields *K*=Q(√*d*) with discriminants 0<*d*<10^{8} and 3-class group of type (3, 3). The results admit extensive statistics of the second 3-class groups G=Gal(F_{3}^{2}*K*∣*K*) and the 3-class field tower groups G=Gal(F_{3}^{∞}*K*∣*K*).

Let p be a prime number. Suppose that K is an algebraic number field with p-class group

degree p, if K possesses the p-class rank

Proposition 1.1. (Order of

The kernel

Proof. The proof of the inclusion

Definition 1.1. For each

If

Corollary 1.1. (Partial and total p-capitulation over

The p-capitulation over K is total if and only if K is real with

Proof. In this special case of a quadratic base field K, the extensions

The organization of this article is the following. In §2, basic theoretical prerequisites for the new capitulation algorithm are developed. The implementation in Magma [

In this article, we consider algebraic number fields K with p-class rank

Definition 2.1. By the Artin pattern of K we understand the pair consisting of the family

Remark 2.1. We usually replace the group objects in the family

We know from Proposition 1.1 that each kernel

Lemma 2.1. Let p be a prime and A be a finite abelian group with positive p-rank and with Sylow p-subgroup

Proof. Any subgroup S of

An application to the particular case

Three cases must be distinguished, according to the abelian type of the p-class group

Lemma 2.2. Let p be a prime number.

Suppose that G is a group and

Then the power

Proof. Generally, the order of a power

This can be seen as follows. Let

Finally, put

Now, we apply Lemma 2.2 to the situation where A is a finite abelian group with type invariants

Proposition 2.1. (p-elementary subgroup)

If A is generated by

Proof. Let generators of A corresponding to the abelian type invariants

Proposition 2.2. (Subgroups of order p)

If the p-elementary subgroup

Proof. According to the assumptions,

selection of generators for the

Proposition 2.3. (Connection between subgroups of index p, resp. order p)

1) If

2) If

3) If

Proof. If

If

If

Theorem 2.1. (Taussky’s conditions A and B, see Formula (5))

Let

Then, we generally have

1) If

L is of type A if either

L is of type B if

2) If

L is of type A if either

L is of type B if

3) If

Proof. This is an immediate consequence of Proposition 2.3. □

Theorem 2.2. (Orbits of TKTs expressing the independence of renumeration)

1) If

2) If

3) If

Proof. The proof for the case

If

In the case

In this section, we present the implementation of our new algorithm for determining the Artin pattern

Algorithm 3.1 (Construction of the base field K and its class group C)

Input: The fundamental discriminant d of a quadratic field

Code:

Output: The conditional class group

Remark 3.1. By using the statement K: =QuadraticField(d); the quadratic field

For the next algorithm it is important to know that in the MAGMA computational algebra system [

Given the situation in Proposition 2.1, where A is a finite abelian group having p-rank

Algorithm 3.2. (Natural ordering of subgroups of index p)

Input: A prime number p and a finite abelian group A with p-rank

Code:

Output: Generators

Proof. This is precisely the implementation of the Propositions 2.1, 2.2 and 2.3 in MAGMA [

Remark 3.2. The modified statement seqS: =Subgroups(A: Quot:=[p,p]); yields the biggest subgroup of A of order coprime to p, and can be used for constructing the Hilbert p-class field

The class group

Algorithm 3.3. (Construction of all unramified cyclic extensions of degree p).

Input: The class group

Code:

Output: Three ordered sequences, seqRelOrd of the relative maximal orders of

Remark 3.3. Algorithm 3.3 is independent of the p-class rank

Algorithm 3.4. (Transfer kernel type,

Input: The prime number p, the ordered sequence seqRelOrd of the relative maximal orders of

Code:

Output: The transfer kernel type TKT of K.

Remark 3.4. In 2012, Bembom investigated the 5-capitulation over complex quadratic fields K with 5-class group of type

Algorithm 3.5. (Transfer target type,

Input: The prime number p and the ordered sequence seqOptAbsOrd of the optimized absolute maximal orders of

Code:

Output: The conditional transfer target type TTT of K, assuming the GRH.

With Algorithms 3.4 and 3.5 we are in the position to determine the Artin pattern

Algorithm 3.6. (Weak transfer kernel type,

Input: The indicators NonCyc, Cyc, and the TKT.

Code:

Output: The weak transfer kernel type TAB of K.

Proof. This is the implementation of Theorem 2.1 in MAGMA [

By means of the algorithms in §3, we have computed the Artin pattern

The 31,088 fields whose second 3-class group

restricted range

451). However, there is a slight increase of 0.37% for the relative frequency of

Theorem 4.1. (Coclass 1) The Hilbert 3-class field tower of a real quadratic field K whose second 3-class group

Proof. This is Theorem 5.3 in [

In

The large scale separation of the types a.2 and a.3, resp. a.2 and a.3, in

Inspired by Boston, Bush and Hajir’s theory of the statistical distribution of p-class tower groups of complex quadratic fields [

Type | AF | RF | MD | |||||
---|---|---|---|---|---|---|---|---|

a.1 | 0000 | 2180 | 7.01% | 62,501 | 3 | |||

a.2 | 1000 | 7104 | 22.85% | 72,329 | 3 | |||

a.3 | 2000 | 10,514 | 33.82% | 32,009 | 3 | |||

a.3^{*} | 2000 | 10,244 | 32.95% | 142,097 | 3 | |||

a.1 | 0000 | 58 | 0.19% | 2,905,160 | 3 | |||

a.2 | 1000 | 242 | 0.78% | 790,085 | 3 | |||

a.3 | 2000 | 713 | 2.29% | 494,236 | 3 | |||

a.1^{2} | 0000 | 3 | 40,980,808 | 3 | ||||

a.2^{2} | 1000 | 9 | 0.03% | 25,714,984 | 3 | |||

a.3^{2} | 2000 | 20 | 0.06% | 10,200,108 | 3 | |||

a.2^{3} | 1000 | 1 | 37,304,664 | 3 | ||||

Total of | 31,088 | 89.77% with respect to 34,631 |

minating types, a.3^{*}, a.3 and a.2.

Conjecture 4.1. For a sufficiently extensive range

Proof. (Attempt of an explanation) A heuristic justification of the conjecture is given for the ground states by the relation for reciprocal orders

which is nearly fulfilled by

For the first excited states, we have the reciprocal orders

but here no arithmetical invariants are known for distinguishing between

have

The 3328 fields whose second 3-class group

([

Theorem 4.2. (Section D) The Hilbert 3-class field tower of a real quadratic field K whose second 3-class group

Proof. This statement has been proved by Scholz and Taussky in ([

^{*} and H.4^{*}, according to ([

Whereas the sufficient criterion for

Theorem 4.3. (Section c) The Hilbert 3-class field tower of a real quadratic field K whose second 3-class group

Proof. This is the union of Thm. 7.1, Cor. 7.1, Cor 7.3, Thm 8.1, Cor 8.1, and Cor 8.3 in [

Type | AF | RF | MD | |||||
---|---|---|---|---|---|---|---|---|

c.18 | 0313 | 347 | 10.4% | 534,824 | 4 | |||

3 | ||||||||

c.21 | 0231 | 358 | 10.8% | 540,365 | 4 | |||

3 | ||||||||

c.18 | 0313 | 8 | 0.2% | 13,714,789 | 4 | |||

c.21 | 0231 | 12 | 0.4% | 1,001,957 | 4 | |||

D.5 | 4224 | 546 | 16.4% | 631,769 | 2 | |||

D.10 | 2241 | 1122 | 33.7% | 422,573 | 2 | |||

E.6 | 1313 | 40 | 1.2% | 5,264,069 | 3 | |||

2 | ||||||||

E.8 | 1231 | 30 | 0.9% | 6,098,360 | 3 | |||

2 | ||||||||

E.9 | 2231 | 83 | 2.5% | 342,664 | 3 | |||

2 | ||||||||

E.14 | 2313 | 63 | 1.9% | 3,918,837 | 3 | |||

2 | ||||||||

E.6 | 1313 | 1 | 75,393,861 | 3 | ||||

E.8 | 1231 | 2 | 26,889,637 | 3 | ||||

E.9 | 2231 | 1 | 79,043,324 | 3 | ||||

E.14 | 2313 | 1 | 70,539,596 | 3 | ||||

G.16 | 4231 | 27 | 0.8% | 8,711,456 | 4 | |||

G.16 | 4231 | 1 | 59,479,964 | 4 | ||||

G.19^{*} | 2143 | 156 | 4.7% | 214,712 | 4 | |||

3 | ||||||||

H.4^{*} | 4443 | 493 | 14.8% | 957,013 | 4 | |||

3 | ||||||||

3 | ||||||||

H.4 | 3313 | 37 | 1.1% | 1,162,949 | 4 | |||

Total of | 3328 | 9.61% with respect to 34,631 |

A sufficient criterion for

Theorem 4.4. (Section E) The Hilbert 3-class field tower of a real quadratic field K whose second 3-class group

Proof. This is the union of Thm. 4.1 and Thm. 4.2 in [

Example 4.1. That both cases

Recently, we have provided evidence of asymptotic frequency distributions for three-stage class field towers, similar to Conjecture 4.1 for two-stage towers.

Conjecture 4.2. For a sufficiently extensive range

Proof. (Attempt of a heuristic justification of the conjecture)

For the first two groups, which form the cover of

which is nearly fulfilled by the statistical information

For the trailing two groups, which form the cover of

Conjecture 4.3. For a sufficiently extensive range

Proof. (Attempt of an explanation) All groups are contained in the cover of

Unfortunately, no arithmetical invariants are known for distinguishing between

There are 190 fields whose second 3-class group

tribution of 0.55%. The corresponding relative frequency for the restricted range

which can be figured out from ([

For the groups

In

We only have 25 fields whose second 3-class group

frequency of

In

a sporadic vertex by

For the essential difference between the location of the groups ^{*} and d.25, see ([

The single occurrence of type H.4 belongs to the irregular variant (i), where

Type | AF | RF | MD | |||||
---|---|---|---|---|---|---|---|---|

b.10 | 0043 | 95 | 50.0% | 710,652 | 5 | |||

b.10 | 0043 | 6 | 3.2% | 17,802,872 | 5 | |||

d.19 | 4043 | 49 | 26.0% | 2,328,721 | 5 | |||

d.23 | 1043 | 16 | 8.4% | 1,535,117 | 5 | |||

d.25 | 2043 | 22 | 12.0% | 15,230,168 | 5 | |||

d.19 | 4043 | 1 | 27,970,737 | 5 | ||||

d.23 | 1043 | 1 | 87,303,181 | 5 | ||||

Total of | 190 | 0.55% with respect to 34,631 |

Type | AF | RF | MD | |||||
---|---|---|---|---|---|---|---|---|

d.25^{*} | 0143 | 4 | 16% | 8,491,713 | 5 | |||

F.7 | 3443 | 3 | 12% | 10,165,597 | 4 | |||

4 | ||||||||

F.11 | 1143 | 3 | 12% | 66,615,244 | 4 | |||

F.12 | 1343 | 6 | 24% | 22,937,941 | 4 | |||

F.13 | 3143 | 5 | 20% | 8,321,505 | 4 | |||

F.7 | 3443 | 1 | 24,138,593 | 4 | ||||

F.12 | 1343 | 1 | 86,865,820 | 4 | ||||

4 | ||||||||

F.13 | 3143 | 1 | 8,127,208 | 4 | ||||

4 | ||||||||

H.4i | 4443 | 1 | 54,313,357 | 4 | ||||

Total of | 25 | 0.07% with respect to 34,631 |

explained in ([

The author gratefully acknowledges that his research is supported by the Austrian Science Fund (FWF): P 26008-N25.

Daniel C. Mayer, (2016) p-Capitulation over Number Fields with p-Class Rank Two. Journal of Applied Mathematics and Physics,04,1280-1293. doi: 10.4236/jamp.2016.47135