Class body tower

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In algebraic number theory , i.e. the theory of complex zeros , of univariate polynomials with whole rational numbers as coefficients , the class field tower of an algebraic number field is understood to be the maximum unbranched pro- extension of for a fixed, given prime number . ${\displaystyle \alpha \in \mathbb {C} ,P(\alpha )=0}$ ${\displaystyle P(X)\in \mathbb {Z} \left[X\right]}$ ${\displaystyle p}$ ${\displaystyle K=\mathbb {Q} (\alpha )}$ ${\displaystyle p}$ ${\displaystyle F_{p}^{\infty }(K)}$ ${\displaystyle K}$ ${\displaystyle p}$

The group of automorphisms of which leave the basic body invariant is called the tower group of . In the case of an infinite class-field tower, there is a topological group with the Krull topology . ${\displaystyle F_{p}^{\infty }(K)}$ ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle G:=G_{p}^{\infty }(K):=Gal(F_{p}^{\infty }(K)/K)}$ ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle F_{p}^{\infty }(K)}$ ${\displaystyle G}$

Table 1: Invariants of the 3-class field towers of imaginary-quadratic number fields

d	type	κ	τ	G 3 2 (K)	G 3 ∞ (K)	Ref.
-3896	H.4	(4443)	[111,111,21,111]	⟨729.45⟩	⟨6561,606⟩
-4027	D.10	(2241)	[21,21,111,21]	⟨243.5⟩	⟨243.5⟩
-9748	E.9	(2231)	[32,21,21,21]	⟨2187,302⟩	⟨6561,620⟩
			or	⟨2187.306⟩	⟨6561,624⟩
-12131	D.5	(4224)	[111,21,111,21]	⟨243.7⟩	⟨243.7⟩
-15544	E.6	(1313)	[32,21,111,21]	⟨2187.288⟩	⟨6561,616⟩
-16627	E.14	(2313)	[32,21,111,21]	⟨2187.289⟩	⟨6561,617⟩
			or	⟨2187.290⟩	⟨6561,618⟩
-34867	E.8	(1231)	[32,21,21,21]	⟨2187,304⟩	⟨6561,622⟩

Levels and length of the p-class field tower

The descending series of iterated commutator groups from , with for , gives rise to the steps (floors, floors) of the tower in the sense of the Galois correspondence , which are given by , or equivalent by , for . ${\displaystyle G}$ ${\displaystyle G=G^{(0)}>G^{(1)}>G^{(2)}>\ldots }$ ${\displaystyle G^{(n+1)}:=\left[G^{(n)},G^{(n)}\right]}$ ${\displaystyle n\geq 0}$ ${\displaystyle F_{p}^{n}(K):=Fix(G^{(n)})}$ ${\displaystyle Gal(F_{p}^{\infty }(K)/F_{p}^{n}(K))=G^{(n)}}$ ${\displaystyle n\geq 0}$

Due to the isomorphism theorem , the Galois group of the -th Hilbert class field of , isomorphic to the -th derived quotient of , and is called the -th class group of . For , with the help of Artin's law of reciprocity, the isomorphism of the Abelianization of the p tower group ,, to the (ordinary) first class group of, i.e. to the Sylow p subgroup of the (finite Abelian) ideal class group of , as the Galois group of the maximum abelian unbranched -extension of . ${\displaystyle G_{p}^{n}(K):=Gal(F_{p}^{n}(K)/K)\cong Gal(F_{p}^{\infty }(K)/K)/Gal(F_{p}^{\infty }(K)/F_{p}^{n}(K))=G/G^{(n)}}$ ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle F_{p}^{n}(K)}$ ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle G}$ ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle K}$ ${\displaystyle n=1}$ ${\displaystyle G/G^{(1)}\cong Gal(F_{p}^{1}(K)/K)\cong Cl_{p}(K)}$ ${\displaystyle p}$ ${\displaystyle K}$ ${\displaystyle Cl(K)}$ ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle F_{p}^{1}(K)}$ ${\displaystyle K}$

The tower group is either an infinite pro group with finite Abelianization or a finite group. In the former case is also the -Klassenkörperturm of , , of infinite length and is the inverse limit of the Galois groups at all levels of the tower. In the latter case is dissolvable and nilpotent and the tower ends at the derived length , expressed more precisely, there is stationary. ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle G/G^{(1)}}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle K}$ ${\displaystyle K<F_{p}^{1}(K)<F_{p}^{2}(K)<\ldots <F_{p}^{\infty }(K)}$ ${\displaystyle \lambda _{p}(K)=\infty }$ ${\displaystyle G=G_{p}^{\infty }(K)\cong \lim _{\leftarrow }G_{p}^{n}(K)}$ ${\displaystyle G}$ ${\displaystyle K<F_{p}^{1}(K)<F_{p}^{2}(K)<\ldots <F_{p}^{\lambda }(K)=F_{p}^{\lambda +1}(K)=F_{p}^{\infty }(K)}$ ${\displaystyle G}$ ${\displaystyle \lambda _{p}(K)=\lambda =dl(G)}$

Relational rank of the p tower group

For the determination of the length of a -class-field tower, the estimation of the relative rank is of decisive importance. ${\displaystyle p}$ ${\displaystyle G=G_{p}^{\infty }(K)}$

${\displaystyle G}$ operates trivially on the finite field with elements and the cohomological dimensions , or , respectively , are called generator rank , or relation rank , of . ${\displaystyle GF(p)}$ ${\displaystyle p}$ ${\displaystyle d_{1}(G):=dimH^{1}(G,GF(p))}$ ${\displaystyle d_{2}(G):=dimH^{2}(G,GF(p))}$ ${\displaystyle G}$

For a basic body with the signature , i.e. with the torsion-free Dirichlet unit rank , Shafarevich has derived the following estimate of the relational rank of the -tower group: where , if the -th contains unit roots , and otherwise. ${\displaystyle K}$ ${\displaystyle (r,c)}$ ${\displaystyle u:=r+c-1}$ ${\displaystyle p}$ ${\displaystyle d_{1}(G)\leq d_{2}(G)\leq d_{1}(G)+u+\theta }$ ${\displaystyle \theta :=1}$ ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle \theta :=0}$

Because of the isomorphism of generator rank of equal to -Klassenrang of , so is the number of basic elements of -Klassengruppe . ${\displaystyle G/G^{(1)}\cong Cl_{p}(K)}$ ${\displaystyle d_{1}(G)}$ ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle \rho _{p}(K)}$ ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle Cl_{p}(K)}$

For the simplest special case of an imaginary-square basic body , which has been particularly extensively investigated , Koch and Venkov derived the following basic result from Shafarevich's cohomological criterion . ${\displaystyle K}$

Theorem of Koch and Venkov. For an odd prime number , the tower group of an imaginary quadratic number field is a so-called Schur σ group with a balanced presentation and an automorphism , which causes the inversion on the Abelianization . (Because of the isomorphism is a generators-inverting (GI) automorphism.) ${\displaystyle p\geq 3}$ ${\displaystyle p}$ ${\displaystyle G=G_{p}^{\infty }(K)}$ ${\displaystyle K}$ ${\displaystyle d_{2}(G)=d_{1}(G)}$ ${\displaystyle \sigma \in Aut(G)}$ ${\displaystyle G/G^{(1)}}$ ${\displaystyle x\rightarrow x^{-1}}$ ${\displaystyle G/G^{(1)}\cong H^{1}(G,GF(p))}$ ${\displaystyle \sigma }$

Addition of Schoof. For an odd prime number and for every integer , the -th class group of any (imaginary or real) square number field has an automorphism that induces both on and on the inversion . ( Therefore it is called a relator-inverting (RI) automorphism.) ${\displaystyle p\geq 3}$ ${\displaystyle n\geq 2}$ ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle U=G_{p}^{n}(K)}$ ${\displaystyle K}$ ${\displaystyle \sigma \in Aut(U)}$ ${\displaystyle H^{1}(U,GF(p))}$ ${\displaystyle H^{2}(U,GF(p))}$ ${\displaystyle x\rightarrow x^{-1}}$ ${\displaystyle \sigma }$

Table 2: Invariants of the 3-class field towers of real-square number fields

d	type	κ	τ	G 3 2 (K)	G 3 ∞ (K)	Ref.
32009	a.3	(2000)	[21,11,11,11]	⟨81.8⟩	⟨81.8⟩
62501	a.1	(0000)	[22,11,11,11]	⟨729.99⟩	⟨729.99⟩
72329	a.2	(1000)	[21,11,11,11]	⟨81.10⟩	⟨81.10⟩
142097	a.3	(2000)	[111,11,11,11]	⟨81.7⟩	⟨81.7⟩
152949	a.1	(0000)	[22,11,11,11]	⟨729,100⟩	⟨729,100⟩
214712	G.19	(4321)	[21,21,21,21]	⟨729.57⟩	⟨2187.311⟩
252977	a.1	(0000)	[22,11,11,11]	⟨729.101⟩	⟨729.101⟩
342664	E.9	(2231)	[32,21,21,21]	⟨2187,302⟩	⟨6561,620⟩
			or	⟨2187.306⟩	⟨6561,624⟩
494236	a.3 ↑	(2000)	[32,11,11,11]	⟨729.97⟩	⟨729.97⟩
			or	⟨729.98⟩	⟨729.98⟩
534824	c.18	(0313)	[22,21,111,21]	⟨729.49⟩	⟨2187.291⟩
540365	c.21	(0231)	[22,21,21,21]	⟨729.54⟩	⟨2187,307⟩
				or	⟨2187.308⟩
790085	a.2 ↑	(1000)	[32,11,11,11]	⟨729.96⟩	⟨729.96⟩
957013	H.4	(4443)	[111,111,21,111]	⟨729.45⟩	⟨2187.273⟩
2905160	a.1 ↑	(0000)	[33,11,11,11]	⟨6561.2227⟩	⟨6561.2227⟩
3918837	E.14	(2313)	[32,21,111,21]	⟨2187.289⟩	⟨2187.289⟩
			or	⟨2187.290⟩	⟨2187.290⟩
4760877	E.9	(2231)	[32,21,21,21]	⟨2187,302⟩	⟨2187,302⟩
			or	⟨2187.306⟩	⟨2187.306⟩
5264069	E.6	(1313)	[32,21,111,21]	⟨2187.288⟩	⟨6561,616⟩
6098360	E.8	(1231)	[32,21,21,21]	⟨2187,304⟩	⟨6561,622⟩
7153097	E.6	(1313)	[32,21,111,21]	⟨2187.288⟩	⟨2187.288⟩
8632716	E.8	(1231)	[32,21,21,21]	⟨2187,304⟩	⟨2187,304⟩
9433849	E.14	(2313)	[32,21,111,21]	⟨2187.289⟩	⟨6561,617⟩
			or	⟨2187.290⟩	⟨6561,618⟩
10200108	a.3 ↑↑	(2000)	[43,11,11,11]	⟨6561.2223⟩	⟨6561.2223⟩
			or	⟨6561.2224⟩	⟨6561.2224⟩
10399596	a.1 ↑	(0000)	[33,11,11,11]	⟨6561.2225⟩	⟨6561.2225⟩
14458876	a.2 ↑↑	(1000)	[43,11,11,11]	⟨6561.2222⟩	⟨6561.2222⟩
27780297	a.1 ↑	(0000)	[33,11,11,11]	⟨6561.2226⟩	⟨6561.2226⟩

Artin pattern of the p tower group

On the way to identify the tower group of a given number field , the Artin pattern is first used to find the metabelianization of . ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle AP=(\kappa ,\tau )}$ ${\displaystyle M:=G/G^{(2)}}$ ${\displaystyle G}$

This pattern consists of the totality of the cores and goals , more precisely: the logarithmic Abelian quotient invariants of the goals, the Artin shifts of the group into its maximum subgroups of the index . ${\displaystyle \kappa :=(ker(V_{G,U}))}$ ${\displaystyle \tau :=(U/U^{(1)})}$ ${\displaystyle V_{G,U}:G/G^{(1)}\to U/U^{(1)}}$ ${\displaystyle G}$ ${\displaystyle U}$ ${\displaystyle p}$

In many cases this strategy of pattern recognition by means of Artin shifts leads to the unambiguous identification of at least the second level of the tower, i.e. the Metabelian Galois group of the second Hilbert class field of , as an approximation of the full tower group . In any case, this process only delivers a finite number of candidates for . ${\displaystyle M\cong G_{p}^{2}(K)}$ ${\displaystyle p}$ ${\displaystyle F_{p}^{2}(K)}$ ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle G_{p}^{2}(K)}$

Historically, the idea for this procedure goes back to the investigations by Arnold Scholz and Olga Taussky-Todd in 1934, from which the designations for the type in Tables 1 and 2 derive. These authors determined the symbolic order from the surrender type (short: type ) of imaginary quadratic number fields with elementary class group of rank, i.e. the annihilator ideal of all bivariate polynomials with the property of the main commutator of the metabelian group of generating rank , i.e. with two generators and . ${\displaystyle \kappa }$ ${\displaystyle K}$ ${\displaystyle 3}$ ${\displaystyle Cl_{3}(K)}$ ${\displaystyle \rho _{3}(K)=2}$ ${\displaystyle f(X,Y)\in \mathbb {Z} [X,Y]}$ ${\displaystyle s^{f(x,y)}=1}$ ${\displaystyle s=\left[y,x\right]}$ ${\displaystyle M:=G_{3}^{2}(K)}$ ${\displaystyle d_{1}(M)=2}$ ${\displaystyle x}$ ${\displaystyle y}$

The second component of the Artin pattern came into play with Scholz and Taussky in a rudimentary form in the form of the class numbers of the four unbranched cyclic-cubic extensions of , but together with it was sufficient for the clear identification of the group . ${\displaystyle \tau }$ ${\displaystyle 3}$ ${\displaystyle K}$ ${\displaystyle \kappa }$ ${\displaystyle M}$

In experimental, computer-aided mathematics, the Artin pattern is used as a search term for database queries either in the SmallGroups library or in extensions of this library, which are constructed using the -group generation algorithm by MF Newman and EA O'Brien. The use of the explicit structure of in the form of the Abelian type invariants of the -class groups (instead of the -class numbers ) of the four unbranched cyclic-cubic extensions of can lead to a considerable restriction of the candidates for the group . ${\displaystyle p}$ ${\displaystyle \tau }$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle K}$ ${\displaystyle M}$

Concrete examples of p-class field towers

So far, the class field towers of square number fields for odd prime numbers have been systematically researched . ${\displaystyle p}$ ${\displaystyle p\geq 3}$

A quadratic fields generated by adjunction one of the two zeros and the polynomial with a fundamental discriminant , where the body of rational numbers. Some basic rules for the length of the class field tower of a quadratic number field can be expressed in terms of the class rank of : ${\displaystyle K=\mathbb {Q} \left({\sqrt {d}}\right)}$ ${\displaystyle {\sqrt {d}}}$ ${\displaystyle -{\sqrt {d}}}$ ${\displaystyle P(X)=X^{2}-d}$ ${\displaystyle d\equiv 0,1({\text{mod}}\,4)}$ ${\displaystyle d\neq 1}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \lambda _{p}(K)}$ ${\displaystyle p}$ ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle \rho _{p}(K)}$ ${\displaystyle K}$

1. The trivial case occurs with any number field if and only if the class number of is not divisible by . ${\displaystyle \lambda _{p}(K)=0}$ ${\displaystyle K}$ ${\displaystyle \rho _{p}(K)=0}$ ${\displaystyle K}$ ${\displaystyle p}$
2. Single-stage towers with are characteristic of cyclic -class groups with square bodies . Unfortunately, this equivalence is lost with other types of basic bodies. So for number fields of the third and fourth degree the condition is still sufficient but in general no longer necessary for . ${\displaystyle \lambda _{p}(K)=1}$ ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle \rho _{p}(K)=1}$ ${\displaystyle K}$ ${\displaystyle \rho _{p}(K)=1}$ ${\displaystyle \lambda _{p}(K)=1}$
3. Koch and Venkov have shown that imaginary-quadratic number fields with at least a three-base class group, i.e. with , also have an infinite class field tower . ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle \rho _{p}(K)\geq 3}$ ${\displaystyle p}$ ${\displaystyle \lambda _{p}(K)=\infty }$
4. The most varied case are the quadratic number fields with class rank , for which all lengths are theoretically possible, but of which so far (as of April 28, 2020) only situations with and rigorously could be proven. ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle \rho _{p}(K)=2}$ ${\displaystyle 2\leq \lambda _{p}(K)\leq \infty }$ ${\displaystyle \lambda _{p}(K)=2}$ ${\displaystyle \lambda _{p}(K)=3}$

The second floor of the class field tower of all quadratic number fields with fundamental discriminators in the area and elementary class group of second place was determined in 2010 in a complex project that took several months of CPU time and its underlying novel algorithms under the keyword surrender type using class group structure were published because the algorithm used by Scholz and Taussky, as well as by Heider and Schmithals, would not have been efficient enough to cope with the 4596 number fields to be analyzed. ${\displaystyle G_{3}^{2}(K)}$ ${\displaystyle 3}$ ${\displaystyle K=\mathbb {Q} \left({\sqrt {d}}\right)}$ ${\displaystyle d}$ ${\displaystyle -10^{6}<d<10^{7}}$ ${\displaystyle 3}$ ${\displaystyle Cl_{3}(K)}$

The third floor of a class field tower, namely every imaginary square number field , with an elementary class group of rank two and Artin pattern with surrender of type E.9 and logarithmic Abelian quotient invariants , was first discovered in the course of history in 2012 by Boston, Bush and Mayer unquestionably identified with precise length after Scholz and Taussky, as well as Heider and Schmithals, had asserted the incorrect two-tier structure . The decisive factor for the proof was the fact that the metabelianization , or , of no Schur σ-group , the -Turm group , or , on the other hand, is very much. ${\displaystyle G_{3}^{3}(K)}$ ${\displaystyle 3}$ ${\displaystyle K=\mathbb {Q} \left({\sqrt {d}}\right)}$ ${\displaystyle d<0}$ ${\displaystyle 3}$ ${\displaystyle Cl_{3}(K)}$ ${\displaystyle AP=(\kappa ,\tau )}$ ${\displaystyle \kappa =(2231)}$ ${\displaystyle \tau =[32,21,21,21]}$ ${\displaystyle \lambda _{3}(K)=3}$ ${\displaystyle \lambda _{3}(K)=2}$ ${\displaystyle M=\langle 2187,302\rangle }$ ${\displaystyle \langle 2187,306\rangle }$ ${\displaystyle G}$ ${\displaystyle 3}$ ${\displaystyle G=\langle 6561,620\rangle }$ ${\displaystyle \langle 6561,624\rangle }$ ${\displaystyle dl(G)=3}$

Further exactly three-level -class-field towers of the types E.6, E.14 and E.8 for imaginary-square bodies and c.18 and c.21 for real-square bodies were discovered in 2015. ${\displaystyle 3}$

In the same year, real-square solids of the types E.6, E.14, E.8 and E.9 were also examined for the length of their -class-field tower , whereby the curiosity turned out that that of Scholz and Taussky was wrong for the imaginary case asserted length is actually allowed in the real case and can be distinguished from the three-stage structure by strictly deterministic criteria. ${\displaystyle 3}$ ${\displaystyle \lambda _{3}(K)=2}$ ${\displaystyle \lambda _{3}(K)=3}$

In 2017, finally, the determination of the fine structure of the real-quadratic number fields , with elementary class group of rank two and Artin pattern with fourfold total capitulation of type a.1 (and any Abelian quotient invariants ) using the so-called deep displacements was successful . ${\displaystyle K=\mathbb {Q} \left({\sqrt {d}}\right)}$ ${\displaystyle d>0}$ ${\displaystyle 3}$ ${\displaystyle Cl_{3}(K)}$ ${\displaystyle AP=(\kappa ,\tau )}$ ${\displaystyle \kappa =(0000)}$ ${\displaystyle \tau }$

Table 1 and 2 respectively show the essential invariants of the class field tower of all imaginary or real square number fields with the minimum discriminant for the respective Artin pattern , if the orders of the two Galois groups and the maximum value of the SmallGroups database do not exceed. Numerous concrete examples with tower groups of a higher logarithmic order are known, but should not be explicitly mentioned here because the designation for these groups with relative identities unfortunately takes up a lot of space and looks confusing at first glance. The symbols or, respectively , behind the type emphasize the first or second stimulus states compared to the basic state , which means variants of with a fixed type . With equality of and is , with difference is and . ${\displaystyle 3}$ ${\displaystyle F_{3}^{\infty }(K)}$ ${\displaystyle K=\left({\sqrt {d}}\right)}$ ${\displaystyle d}$ ${\displaystyle AP=(\kappa ,\tau )}$ ${\displaystyle G_{3}^{2}(K)}$ ${\displaystyle G_{3}^{\infty }(K)}$ ${\displaystyle 6561}$ ${\displaystyle 3}$ ${\displaystyle G}$ ${\displaystyle lo(G)>8}$ ${\displaystyle \uparrow }$ ${\displaystyle \uparrow \uparrow }$ ${\displaystyle \tau }$ ${\displaystyle \kappa }$ ${\displaystyle G_{3}^{2}(K)}$ ${\displaystyle G_{3}^{\infty }(K)}$ ${\displaystyle \lambda _{3}(K)=2}$ ${\displaystyle G_{3}^{\infty }(K)=G_{3}^{3}(K)}$ ${\displaystyle \lambda _{3}(K)=3}$

A striking difference in the order of the -tower group and occasionally even in the length of the -class field tower was found between imaginary-quadratic number fields with negative discriminants and real-quadratic number fields with positive discriminants with the same Artin pattern . Thus, the bodies have with discriminant and consistent with the type H.4 , isomorphic second three-class groups and the same length , but the Virgin group in the imaginary case has greater order than the -Turmgruppe with relative rank in the real case. The different behavior of the bodies with discriminants and is even more serious . While type E.6 corresponds to , and the second class groups , the demanding imaginary body, according to the theorem of Koch and Venkov, naturally consists of the Schur group with, but the frugal real body is already satisfied with the unbalanced group with . ${\displaystyle ord(G)}$ ${\displaystyle 3}$ ${\displaystyle G}$ ${\displaystyle \lambda _{3}(K)}$ ${\displaystyle 3}$ ${\displaystyle d<0}$ ${\displaystyle d>0}$ ${\displaystyle AP=(\kappa ,\tau )}$ ${\displaystyle K}$ ${\displaystyle d=-3896}$ ${\displaystyle d=957013}$ ${\displaystyle \kappa =(4443)}$ ${\displaystyle \tau =[111,111,21,111]}$ ${\displaystyle G_{3}^{2}(K)\cong \langle 729,45\rangle }$ ${\displaystyle \lambda _{3}(K)=3}$ ${\displaystyle \sigma }$ ${\displaystyle G_{3}^{\infty }(K)\cong \langle 6561,606\rangle }$ ${\displaystyle 3}$ ${\displaystyle G:=G_{3}^{\infty }(K)\cong \langle 2187,273\rangle }$ ${\displaystyle d_{2}(G)=3}$ ${\displaystyle K}$ ${\displaystyle d=-15544}$ ${\displaystyle d=7153097}$ ${\displaystyle \kappa =(1313)}$ ${\displaystyle \tau =[32,21,111,21]}$ ${\displaystyle 3}$ ${\displaystyle G_{3}^{2}(K)\cong \langle 2187,288\rangle }$ ${\displaystyle \sigma }$ ${\displaystyle G_{3}^{\infty }(K)\cong \langle 6561,616\rangle }$ ${\displaystyle \lambda _{3}(K)=3}$ ${\displaystyle G_{3}^{\infty }(K)\cong \langle 2187,288\rangle }$ ${\displaystyle \lambda _{3}(K)=2}$

Unsolved problems with infinite p-class field tower

The class field tower of a number field can only be considered to be known if there is a pro- presentation of its Galois group, i.e. the tower group , with explicit generators and relations. In the case of an infinite tower with the length , an analytical expression for the growth of the orders of the countably infinite number of steps of the tower as a function of could be given from this pro- presentation . ${\displaystyle p}$ ${\displaystyle F_{p}^{\infty }(K)}$ ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle G_{p}^{\infty }(K)}$ ${\displaystyle \lambda _{p}(K)=\infty }$ ${\displaystyle p}$ ${\displaystyle n\rightarrow Ord_{n}}$ ${\displaystyle Ord_{n}:=ord(G_{p}^{n}(K))}$ ${\displaystyle n\geq 2}$

Unfortunately, the solution to this very interesting problem is still a long way off. For example, the imaginary quadratic number field with fundamental discriminant has an elementary class group of rank , i.e. with logarithmic Abelian type invariants , and thus an infinite class field tower according to the above basic rule 3 . But the values ​​of the function are completely unknown for and for , i.e. for the order of the second class group , only the lower estimate could be calculated with enormous expenditure of CPU time . ${\displaystyle K=\mathbb {Q} ({\sqrt {d}})}$ ${\displaystyle d=-4447704}$ ${\displaystyle 3}$ ${\displaystyle \rho _{3}(K)=3}$ ${\displaystyle [111]}$ ${\displaystyle 3}$ ${\displaystyle \lambda _{3}(K)=\infty }$ ${\displaystyle Ord_{n}}$ ${\displaystyle n\geq 3}$ ${\displaystyle n=2}$ ${\displaystyle 3}$ ${\displaystyle Gal(F_{3}^{2}(K)/K)}$ ${\displaystyle Ord_{2}\geq 3^{17}}$

Individual evidence

1. ↑ ^{a b c d e} D. C. Mayer: Index-p abelianization data of p-class tower groups . In: Adv Pure Math.. . 5, 2015, pp. 286-313. arxiv : 1502.03388 . doi : 10.4236 / apm.2015.55029 .
2. ↑ ^{a b c d e f g h} D. C. Mayer: Principalization algorithm via class group structure . In: J. Théor. Nombres Bordeaux . 26, 2014, pp. 415-464. arxiv : 1403.3839 . doi : 10.5802 / jtnb.874 .
3. ↑ ^{a b} M. R. Bush, DC Mayer: 3-class field towers of exact length 3 . In: J. Number Theory . 147, 2015, pp. 766-777. arxiv : 1312.0251 . doi : 10.1016 / j.jnt.2014.08.010 .
4. ^ E. Artin: Proof of the general law of reciprocity . In: Abh. Math. Sem. Univ. Hamburg . 5, 1927, pp. 353-363.
5. ^ IR Shafarevich: Extensions with given points of ramification . In: Inst. Hautes Études Sci. Publ. Math. . 18, 1963, pp. 71-95. Translated in Amer. Math. Soc. Transl. (2) , 59 : 128-149, (1966).
6. ↑ ^{a b} H. Koch, BB Venkov: About the p-class field tower of an imaginary-quadratic number field . In: Astérisque . 24-25, 1975, pp. 57-67.
7. ↑ René Schoof : Infinite class field towers of quadratic fields . In: J. Reine Angew. Math. . 372, 1986, pp. 209-220.
8. ↑ ^{a b c d e f g h i} D. C. Mayer: Deep transfers of p-class tower groups . In: J. Appl. Math. Phys. . 6, 2018, pp. 36-50. arxiv : 1707.00232 . doi : 10.4236 / jamp.2018.61005 .
9. ↑ ^{a b c d e f g h i j k l m} D. C. Mayer: Criteria for three-stage towers of p-class fields . In: Adv Pure Math.. . 7, 2017, pp. 135-179. arxiv : 1601.00179 . doi : 10.4236 / apm.2017.72008 .
10. ↑ ^{a b c d} D. C. Mayer: New number fields with known p-class tower . In: Tatra Mt. Math. Pub. . 64, 2015, pp. 21-57. arxiv : 1510.00565 . doi : 10.1515 / tmmp-2015-0040 .
11. ↑ ^{a b c} A. Scholz, O. Taussky: The main ideals of the cubic class fields of imaginary quadratic number fields: their computational determination and their influence on the class field tower . In: J. Reine Angew. Math. . 171, 1934, pp. 19-41.
12. ^ DC Mayer: Transfers of metabelian p-groups . In: monthly Math. . 166, 2012, pp. 467-495. arxiv : 1403.3896 . doi : 10.1007 / s00605-010-0277-x .
13. ^ DC Mayer: Annihilator ideals of two-generated metabelian p-groups . In: J. Algebra Appl. . 17, 2018. arxiv : 1603.09288 . doi : 10.1142 / S0219498818500767 .
14. ^ ^{A b} H. U. Besche, B. Eick, EA O'Brien: The SmallGroups Library - a library of groups of small order . An accepted and refereed GAP 4 package, also available in MAGMA, 2005.
15. ↑ MF Newman, Determination of groups of prime-power order, in: Group Theory, Canberra 1975, Springer, Lecture Notes in Mathematics 573, 1977, pp. 73-84
16. ^ EA O'Brien: The p-group generation algorithm . In: J. Symbolic Comput. . 9, 1990, pp. 677-698. doi : 10.1016 / s0747-7171 (08) 80082-x .
17. ^ DC Mayer: The second p-class group of a number field . In: Int. J. Number Theory . 8, 2012, pp. 471-505. arxiv : 1403.3899 . doi : 10.1142 / s179304211250025x .
18. ↑ ^{a b} F.-P. Heider, B. Schmithals: On the surrender of the ideal classes in unbranched primary cyclic extensions . In: J. Reine Angew. Math. . 363, 1982, pp. 1-25.