Principalization (algebra)

This conjecture, now known as principal ideal theorem, was proved by Philipp Furtwängler in 1930 after it had been translated from number theory to group theory by Emil Artin in 1929, who made use of his general reciprocity law to establish the reformulation.

The first contributions in this direction are due to Arnold Scholz and Olga Taussky in 1934, who coined the synonym capitulation for principalization.

Another independent access to the principalization problem via Galois cohomology of unit groups is also due to Hilbert and goes back to the chapter on cyclic extensions of number fields of prime degree in his number report, which culminates in the famous Theorem 94.

is a formal product of pair-wise different real infinite primes of

induces the extension homomorphism of generalized ideal class groups:

which associates an entire conjugacy class of automorphisms to any unramified prime ideal

[4] This reciprocity law allowed Artin to translate the general principalization problem for number fields

This product expression was the original form of the Artin transfer homomorphism, corresponding to a decomposition of the permutation representation into disjoint cycles.

are connected by the commutative diagram in Figure 1 via the isomorphisms induced by the Artin maps, that is, we have equality of two composita

[3][6] The commutative diagram in the previous section, which connects the number theoretic class extension homomorphism

, enabled Furtwängler to prove the principal ideal theorem by specializing to the situation that

However, the commutative diagram comprises the potential for a lot of more sophisticated applications.

via pattern recognition, and frequently even allows to draw conclusions about the entire p-class field tower of

[7] At these early stages, pattern recognition consisted of specifying the annihilator ideals, or symbolic orders, and the Schreier relations of metabelian p-groups and subsequently using a uniqueness theorem on group extensions by O.

[8] Nowadays, we use the p-group generation algorithm of M. F. Newman[9] and E. A. O'Brien[10] for constructing descendant trees of p-groups and searching patterns, defined by kernels and targets of Artin transfers, among the vertices of these trees.

In the chapter on cyclic extensions of number fields of prime degree of his number report from 1897, D. Hilbert[2] proves a series of crucial theorems which culminate in Theorem 94, the original germ of class field theory.

of the extension field, viewed as a Galois module with respect to the group

In fact, the image of the algebraic norm map is contained in the unit group

Two important cohomology groups can be defined by means of the kernels and images of these endomorphisms.

is a cyclic extension of number fields of odd prime degree

, which means it's unramified at finite primes, then there exists a non-principal ideal

As mentioned in the lead section, several investigators tried to generalize the Hilbert-Artin-Furtwängler principal ideal theorem of 1930 to questions concerning the principalization in intermediate extensions between the base field and its Hilbert class field.

On the one hand, they established general theorems on the principalization over arbitrary number fields, such as Ph.

[16] On the other hand, they searched for concrete numerical examples of principalization in unramified cyclic extensions of particular kinds of base fields.

-class rank two in unramified cyclic cubic extensions was calculated manually for three discriminants

Since these calculations require composition of binary quadratic forms and explicit knowledge of fundamental systems of units in cubic number fields, which was a very difficult task in 1934, the investigations stayed at rest for half a century until F.-P. Heider and B. Schmithals[17] employed the CDC Cyber 76 computer at the University of Cologne to extend the information concerning principalization to the range

relevant discriminants in 1982, thereby providing the first analysis of five real quadratic fields.

[21] Similar investigations of real quadratic fields were carried out by E. Benjamin and C. Snyder in 1995.

-principalization in unramified quadratic extensions of biquadratic fields of Dirichlet type

-principalization in the two layers of unramified quadratic and biquadratic extensions of quartic fields with class groups of