Kenkichi Iwasawa

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Kenkichi Iwasawa (1917–1998) was a Japanese mathematician renowned for his foundational work in algebraic number theory. He developed what is now called Iwasawa theory, which explores the deep interplay between p-adic analysis, class groups, and infinite Galois extensions. His contributions have had a lasting impact on modern number theory, influencing the study of p-adic L-functions and the arithmetic of cyclotomic fields. Iwasawa served on the faculty at institutions such as Princeton University before returning to Japan, where he continued his research and mentorship.

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All published works (43)

Algebraic Number Fields

2019-01-01

Kenkichi Iwasawa

Hecke’s L-functions

2019-01-01

Kenkichi Iwasawa

L-functions

2019-01-01

Kenkichi Iwasawa

Erratum to: On the paper “On the group of automorphisms of a function field”

2001-01-01

Kenkichi Iwasawa , Tsuneo Tamagawa

A note on cyclotomic fields

2001-01-01

Kenkichi Iwasawa

Analytic theory of algebraic function fields

1993-04-20

Kenkichi Iwasawa

Algebraic function fields and closed Riemann surfaces

1993-04-20

Kenkichi Iwasawa

Algebraic functions

1993-04-20

Kenkichi Iwasawa

A note on capitulation problem for number fields, II

1989-01-01

Kenkichi Iwasawa

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A note on capitulation problem for number fields

1989-01-01

Kenkichi Iwasawa

Let F be a finite extension of a finite algebraic number field k and let C and C denote the ideal class groups of k and F respectively.A sub- group … Let F be a finite extension of a finite algebraic number field k and let C and C denote the ideal class groups of k and F respectively.A sub- group A of C is said to capitulates in F if A-+I under the natural homo- morphism C--+C.The principal ideal theorem of class field theory states that C always capitulates in Hilbert's class field K over k.How- ever, as shown in Heider-Schmithals [1], for some k, C capitulates already in a proper subfield M of K" k_M_K, M=/=K.In the present note, we shall give further simple examples of such number fields k for which the capitulation of C occurs in a proper subfield M of Hilbert's class field K over k*).1. Let L be a finite abelin (or nilpotent) extension over k.For each

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On Cohomology Groups of Units for Z p -Extensions

1983-02-01

Kenkichi Iwasawa

Let K/k be a Galois extension of number fields, S a set of finite places on the ground field k, and Es the group of S-units in K. The cohomology … Let K/k be a Galois extension of number fields, S a set of finite places on the ground field k, and Es the group of S-units in K. The cohomology groups Hn(K/k, Es), i.e., Hn(Gal(K/k), Es), have been studied for various K/k and S, and found to be an important device to study the arithmetic properties of the extension K/k. In the present paper, we shall make some simple remarks on H'(K/k, Es) in the special case where K is a so-called Zp-extension over a finite algebraic number field k.

Riemann-Hurwitz formula and $p$-adic Galois representations for number fields

1981-01-01

Kenkichi Iwasawa

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On P-ADIC Representations Associated with ℤp -Extensions

1981-01-01

Kenkichi Iwasawa

A note on cyclotomic fields

1976-12-01

Kenkichi Iwasawa

On Z l -Extensions of Algebraic Number Fields

1973-09-01

Kenkichi Iwasawa

On the µ-invariants of cyclotomic fields

1972-01-01

Kenkichi Iwasawa

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Skew-symmetric forms for number fields

1971-01-01

Kenkichi Iwasawa

On p-Adic L-Functions

1969-01-01

Kenkichi Iwasawa

A Note on Ideal Class Groups

1966-02-01

Kenkichi Iwasawa

In the first part of the present paper, we shall make some simple observations on the ideal class groups of algebraic number fields, following the group-theoretical method of Tschebotarew. The … In the first part of the present paper, we shall make some simple observations on the ideal class groups of algebraic number fields, following the group-theoretical method of Tschebotarew. The applications on cyclotomic fields (Theorems 5, 6) may be of some interest. In the last section, we shall give a proof to a theorem of Kummer on the ideal class group of a cyclotomic field.

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Seminar on Complex Multiplication

1966-01-01

A. Borel , S. Chowla , C. S. Herz +2 more

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Computation of invariants in the theory of cyclotomic fields

1966-01-01

Kenkichi Iwasawa , Charles C. Sims

1. Let a prime number $p$ be fixed, and let $F_{n},$ $n\geqq 0$ , denote the cyclotomic field of $p^{n+1}$ -th roots of unity over the rational field $Q$ .Let … 1. Let a prime number $p$ be fixed, and let $F_{n},$ $n\geqq 0$ , denote the cyclotomic field of $p^{n+1}$ -th roots of unity over the rational field $Q$ .Let $p^{c(n)}$ be the highest power of $p$ dividing the class number $h_{n}$ of $F_{n}$ .Then there exist integers $\lambda_{p}$ , $\mu_{p}$ , and $\nu_{p}(\lambda_{p}, \mu_{p}\geqq 0)$ , depending only upon $p$ , such that $c(n)=\lambda_{p}n+\mu_{p}p^{n}+\nu_{p}$ , for every sufficiently large integer $n^{1)}$ .In the present paper, we shall deter- mine, by the help of a computer, the coefficients $\lambda_{p},$ $\mu_{p}$ , and $1_{I)}^{1}$ in the above formula for all prime numbers $p\leqq 4001$ .We shall see in particular that $\mu_{p}=0$ for every $p\leqq 4001$ .Let $S_{n}$ denote the Sylow p-subgroup of the ideal class group of $F_{n}$ .For the above primes, we shall determine not only the order $p^{c(n)}$ of $S_{n}$ but also the structure of the abelian group $S_{n}$ for every $n\geqq 0$ .Let $p=2$.Then we know by Weber's theorem that $c(n)=0,$ $S_{n}=1$ for any $n\geqq 0$ so that $\lambda_{2}=\mu_{2}=\nu_{2}=0$ .Therefore, we shall assume throughout the following that $p$ is an odd prime, $p>2$ .2. Let $Q_{p}$ and $Z_{p}$ denote the field of p-adic numbers and the ring of $p$ -adic integers, respectively.Let $F$ be the union of all fields $F_{n},$ $n\geqq 0$ .Then $F$ is an abelian extension of $Q$ , and we denote the Galois group of $F/Q$ by $G$ .For each p-adic unit $u$ in $Q_{p}$ , there is a unique automorphism $\sigma_{u}$ of $F$ such that $\sigma_{u}(\zeta)=\zeta^{u}$ for any root of unity $\zeta$ in $F$ with order a power of $p$ .The mapping $u\rightarrow\sigma_{u}$ then defines a topological isomorphism of the group of $p$ -adic units in $Q_{p}$ onto the compact abelian group $G$ .Let $\Gamma$ and $\Delta$ denote the sub- groups of $G$ corresponding to the group of l-units in $Q_{p}$ and the group $V$ of all $(p-1)-st$ roots of unity in $Q_{p}$ , respectively.Then we have $ G=\Gamma\times\Delta$ ;

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Some results in the theory of cyclotomic fields

1965-01-01

Kenkichi Iwasawa

On some modules in the theory of cyclotomic fields

1964-01-01

Kenkichi Iwasawa

On some modules in the theory of cyclotomic fields by Kenkichi IWASAWA On some modules in the theory of cyclotomic fields by Kenkichi IWASAWA

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A Class Number Formula for Cyclotomic Fields

1962-07-01

Kenkichi Iwasawa

where En+ is the group of units in F,+ and To is the subgroup of circular units in E.+.' Let Gn denote the Galois group of Fn over Q and … where En+ is the group of units in F,+ and To is the subgroup of circular units in E.+.' Let Gn denote the Galois group of Fn over Q and let 9R = Z[Gn] be the group ring of Gn over the ring of rational integers Z. In the present paper, we shall first transform the formula (1) and show that the first factor halso can be expressed as a group index of certain additive groups in R. In general, any such class number formula suggests the existence of deeper group-theoretical relations between the ideal class group and the factor group by the order of which the class number is expressed.2 In the rest of the paper, we shall study such relations between the two groups involved in our formula for he-.

On local cyclotomic fields.

1960-01-01

Kenkichi Iwasawa

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On the Theory of Cyclotomic Fields

1959-11-01

Kenkichi Iwasawa

Sheaves for Algebraic Number Fields

1959-03-01

Kenkichi Iwasawa

On Γ-extensions of algebraic number fields

1959-01-01

Kenkichi Iwasawa

KENKICHI IWASAWA 1Let p be a prime number.We call a Galois extension L of a field K a T-extension when its Galois group is topologically isomorphic with the additive group … KENKICHI IWASAWA 1Let p be a prime number.We call a Galois extension L of a field K a T-extension when its Galois group is topologically isomorphic with the additive group of £-adic integers.The purpose of the present paper is to study arithmetic properties of such a T-extension L over a finite algebraic number field K.We consider, namely, the maximal unramified abelian ^-extension M over L and study the structure of the Galois group G(M/L) of the extension M/L.Using the result thus obtained for the group G(M/L)> we then define two invariants l(L/K) and m(L/K) } and show that these invariants can be also determined from a simple formula which gives the exponents of the ^-powers in the class numbers of the intermediate fields of K and L. Thus, giving a relation between the structure of the Galois group of M/L and the class numbers of the subfields of L, our result may be regarded, in a sense, as an analogue, for L, of the well-known theorem in classical class field theory which states that the class number of a finite algebraic number field is equal to the degree of the maximal unramified abelian extension over that field.An outline of the paper is as follows: in §1- §5, we study the structure of what we call T-finite modules and find, in particular, invariants of such modules which are similar to the invariants of finite abelian groups.In §6, we give some definitions and simple results on certain extensions of (infinite) algebraic number fields, making it clear what we mean by, e.g., an unramified extension, when the ground field is an infinite algebraic number field.In the last §7, we first show that the Galois group G(M/L) as considered above is a T-finite module, then define the invariants l(L/K) and rn(L/K), and finally prove our main formula using the group-theoretical results obtained in previous sections.1. Preliminaries.1.1.Let p be a prime number.We shall first recall some definitions and elementary properties of ^-primary abelian groups.

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Book Review: Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlkörpern

1958-07-01

Kenkichi Iwasawa

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On Some Invariants of Cyclotomic Fields

1958-07-01

Kenkichi Iwasawa

where A, u and v are integers independent of n, The numbers A and a seem to have deep significance for the arithmetic of the fields Kn. In general, if … where A, u and v are integers independent of n, The numbers A and a seem to have deep significance for the arithmetic of the fields Kn. In general, if the invariant 1 ,u(Kf/F) of a so-called r-extension K over a finite algebraic number field F is 0, then the Galois group of the maximal unramified abelian p-extension over K is, up to a finite subgroup, isomorphic with the direct sum of A copies of the additive group of p-adic integers, where A-= X(K/F) denotes another invariant of K/F.1 So, if / 0, we have an analogue, for number fields, of a similar result for algebraic function fields of one variable over algebraically closed fields of constants. For this and other reasons, it seems interesting to know whether a> 0 or , = 0 for a given I-extension K/F, and we shall find ill the present paper necessary aind sufficient conditions for y > 0 when the I-extension is obtained from the cyclotomic fields Kn defined above, namely, when u is given as the second coefficient in the above formula (1).

A note on class numbers of algebraic number fields

1956-03-01

Kenkichi Iwasawa

On Galois Groups of Local Fields

1955-11-01

Kenkichi Iwasawa

Let p be a prime number, Qp the field of p-adic numbers, and fl an algebraic closure of Qp.In the present paper, we take a finite extension k of Qv … Let p be a prime number, Qp the field of p-adic numbers, and fl an algebraic closure of Qp.In the present paper, we take a finite extension k of Qv in ft as the ground field and study the structure of the Galois group G(Q/k) of the extension Q./k.Let V he the ramification field of Q/k, i.e. the composite of all finite tamely ramified extensions of k in Q, and let G(&/ V) and G( V/k) denote the Galois groups of the extensions Q/F and V/k respectively.We shall first determine the structure of the groups G(V/k) =G(£l/k)/G(ti/V), and G(Q,/V) and show that the group extension G(Q/k)/G(Q/V) splits.Our main result is, then, to describe explicitly the effect of inner automorphisms of G(Q./k) on the factor group of G(Q/V) modulo its commutator subgroup, i.e., on the Galois group G(V'/V) of the maximal abelian extension V of V in ft.This is, of course, not sufficient to determine the structure of the group G(Sl/k) completely; to do that, we still have to find the effect of inner automorphisms of G(ti/k) on the normal subgroup G(Q/V) itself.However, it gives us some insight into the structure of G(il/k); and we hope it will help somehow, in the future, in the study of the group G(ft/&) as well as in that of the Galois groups of algebraic number fields.An outline of the paper is as follows: in §1 we prove some group-theoretical lemmas which will be used later.In §2 we study the behavior of the Galois group of a certain type of finite tamely ramified Galois extension E oi k acting on the multiplicative group of E. Using those results, we then prove in §3 the properties of G(Sl/k) as mentioned above(1).

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On Galois groups of local fields

1955-01-01

Kenkichi Iwasawa

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A note on Kummer extensions

1953-11-01

Kenkichi Iwasawa

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On Solvable Extentions of Algebraic Number Fields

1953-11-01

Kenkichi Iwasawa

On the Rings of Valuation Vectors

1953-03-01

Kenkichi Iwasawa

Let {R,,,} be a system of topological rings and Qa open subrings of Ra . We consider the set R of all vectors a = (aa), where aa are elements … Let {R,,,} be a system of topological rings and Qa open subrings of Ra . We consider the set R of all vectors a = (aa), where aa are elements in Ra and which belong to Qa, except for a finite number of a. By the usual definition of component-wise addition and multiplication, R forms a ring containing the direct sum 0 of all Qa . We can then define, in a unique manner, a topology in R so that R becomes a topological ring, 0 becomes an open subring of R and is the Cartesian product of Qa as a topological space. We call R the local direct sum of Ra relative to O. .1 Now, let k be a finite algebraic number field or an algebraic function field of one variable over a constant field F. In the following, we shall call such a field, simply, a number field or a function field respectively. We consider the set {Kp} of all completions of K with respect to prime divisors P of K, which are trivial on F in the case of a function field, and denote by Op the valuation ring of P in Kp, if P is non-archimedean, and the field Kp itself, if P is archimedean. Then, with respect to the usual topology of Kp induced by a valuation of P, each Op is an open subring of Kp , and we can form the local direct sum R of Kp relative to Op . We call this R the ring of valuation vectors of K. If we identify each element E of K with the vector a = (aa), whose components aa are all equal to b, K is isomorphically imbedded in R and becomes a discrete subfield of R, as we shall show later. According to recent results in algebraic number theory, it has become clearer and clearer that the topological properties of R and of its related structures, in particular that of the multiplicative group of R, have essential relations to the arithmetic of the field K.2 Hence, it seems to be of some interest to know what are the characteristic properties of R as a topological ring, for this would show us the sources of arithmetic theorems which can be deduced from the topological structure of R, and might possibly give us some suggestions for further developments in algebraic number theory. The main purpose of the present paper is to give such a characterization of the rings of valuation vectors of number fields and function fields. After some preparations in ?1, we shall do this in ?2 and ?3, and then show in ?4, by some examples, how arithmetic properties of K are related to the topological structure of the ring R. For the proofs, we shall use Haar measures of locally compact groups, some fundamental properties of locally compact rings and, in particular,

Correction: On the paper `` On the group of automorphisms of a function field ''.

1952-10-01

Kenkichi Iwasawa , Tsuneo Tamagawa

A correction was made to our paper mentioned in the title, as the proof of Lemma 1 was not complete.We give here another correction, which will give more explicitly the … A correction was made to our paper mentioned in the title, as the proof of Lemma 1 was not complete.We give here another correction, which will give more explicitly the bound of the order of the automorphism in question.We use the same notations as in Lemma 1 of the original paper, without mentioning explicitly their meanings.Here we are concerned with the modular case, $i$ .$e$ .the case where the characteristic $p$ of $k$ is $\neq 0$ .Obviously we can assume either $\sigma(x)=x+\alpha$ or $\sigma(x)=\alpha x$ with some $\alpha$ in $k$ .In the first case, the order of $\sigma$ does not exceed $pn$ .Hence we assume $\sigma(x)=\alpha x$ .If the divisor of $x$ is of the form $P^{n}Q^{-n}$ , where $P,$ $Q$ are completely ramified primes of $K$ over $K^{\prime}$ , the contributions of $P$ and $Q$ to the different of $K/K^{\prime}$ are $P^{n-1}$ and $Q^{n-1}$ respectively.The original proof of Lemma 1 can be applied to this case and we see that the order of $\sigma$ is at most $n(2n+2g-2)(2n+2g-3)(2n+2g-4)$ .Suppose, now, that either the numerator or the denominator of $x$ , say the latter, contains two different primes $P_{1},$ $P_{2}$ of $K$ A suitable power $\tau=\sigma^{l},$ $l\leqq n$ , leaves $P_{1}$ fixed, and we can find an element $y$ in $K$ such that $\tau(y)=\beta y+\gamma,$ $\beta,$ $\gamma\in k$ , and that the denominator of $y$ is $P_{1}^{r},$ $r\leqq g+1$ (cf. the proof in p. 139 of the original paper).If $\beta=1$ , the order of $\tau$ is at most $p(g+1)$ and, consequently, the order of $\sigma$ is at most $pn(g+1)$ .We may therefore assume that $\beta\neq 1$ and $\tau(y)=\beta y$ .Let $F(X, Y)=\sum\alpha_{ij}X^{i}Y^{j}$ be an irreducible polynomial over $k$ such that $F(x,y)=0$ .Since $\tau(x)=\alpha^{l}x,$ $\tau(y)=\beta y$ , we have $F(\alpha^{l}X, \beta Y)=$ $\xi F(X, Y),$ $\xi\in k$ .Therefore, if $\alpha_{ij}\neq 0,$ $\alpha_{sl}\neq 0,$ $(i,j)\neq(s, t),$ $z=x^{i-s}y^{j-t}$

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Topological Groups With Invariant Compact Neighborhoods of the Identity

1951-09-01

Kenkichi Iwasawa

On the Group of Automorphisms of a Function Field

1951-05-01

Kenkichi Iwasawa , Tsuneo Tamagawa

\S 1.Let lf be an algebraic function field over an algebraically closed constant field $k$ .It is well-known that the group of automorphisms of $K$ over $k$ is a finite … \S 1.Let lf be an algebraic function field over an algebraically closed constant field $k$ .It is well-known that the group of automorphisms of $K$ over $k$ is a finite group, when the genus of $K$ is greater than 1.In the classical case, where $k$ is the field of complex numbers, this theorem was proved by Klein and Poincar\'e1) by making use of the analytic theory of Riemann surfaces.On the other hand, Weierstrass and Hurwitz gave more aigebraical proofs in the same $case^{\underline{o}}$ ) $whi_{-}^{\backslash }h$ essentially depend upon the existence of $c_{O}$ -called Weierstrass points of $K$ .Because of its algebraic nature, the latter method is immediately applicable to the case of an arbitrary constant field of characteristic zero.In the case of characteristic $p\neq 0$ , H. L. $S_{\sim}\wedge hmidP^{loved}$ the theorem along similar lines; the proof being based upon F. K. Schmidt's generalization of the classical theory of Weierstrass points in such a case4).Now it has been remarked, since Hurwitz, that the representation of $tlle$ group $G$ of automo;phisms of $K$ by the linear $trallsfolmations,$ $i_{1^{\neg}}.duced$ by $G$ in the set of all differentials of the first kind of $K$ , is very impor-

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On Some Types of Topological Groups

1949-07-01

Kenkichi Iwasawa

As is well-known the so-called fifth problem of Hilbert on continuous groups was solved by J. v. Neumann [14]2 for compact groups and by L. Pontrjagin [15] for abelian groups. … As is well-known the so-called fifth problem of Hilbert on continuous groups was solved by J. v. Neumann [14]2 for compact groups and by L. Pontrjagin [15] for abelian groups. More recently, it is reported, C. Chevalley [6] solved it for solvable groups.3 Now it seems, as H. Freudenthal [7] clarified for maximally almost periodic groups, that the essential source of the proof of Hilbert's problem for these groups lies in the fact that such groups can be approximated by Lie groups. Here we say that a locally compact group G can be approximated by Lie groups, if G contains a system of normal subgroups {Na} such that G/Na are Lie groups and that the intersection of all Na coincides with the identity e. For the brevity we call such a group a group of type (L) or an (L)-group. In the present paper we shall study the structure of such (L)-groups, and apply the result to solve the Hilbert's problem for a certain class of groups, which contains both compact and solvable groups as special cases. We shall be able to characterize a Lie group G, for which the factor group GIN of G modulo its radical N is compact, completely by its structure as a topological group. The outline of the paper is as follows. In ?1 we study the topological structure of the group of automorphisms of a compact group and prove theorems concerning compact normal subgroups of a connected topological group, which are to be used repeatedly in succeeding sections. In ?2 come some preliminary considerations on solvable groups, whereas finer structural theorems on these groups are, as special cases of (L)-groups, given later. In ?3 we prove some theorems on Lie groups. The theorems here stated are not all new, but we give them here for the sake of completeness, and thereby refine and modify these theorems so as to be applied appropriately in succeeding sections.4 After these preparations we study in ?4 the structure of (L)-groups. In particular, it is shown 'that the study of the local structure and the global topological structure

On linearly ordered groups.

1948-09-01

Kenkichi Iwasawa

On linearly ordered groups. On linearly ordered groups.

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On group rings of topological groups

1944-01-01

Kenkichi Iwasawa

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On the Structure of Conditionally Complete Lattice-Groups

1941-01-01

Kenkichi Iwasawa

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Common Coauthors

Coauthor	Papers Together
Tsuneo Tamagawa	3
S. Chowla	1
C. S. Herz	1
A. Borel	1
J-P. Serre	1
Charles C. Sims	1

Commonly Cited References

On the Theory of Cyclotomic Fields

1959-11-01

Kenkichi Iwasawa

On Z l -Extensions of Algebraic Number Fields

1973-09-01

Kenkichi Iwasawa

On some modules in the theory of cyclotomic fields

1964-01-01

Kenkichi Iwasawa

On some modules in the theory of cyclotomic fields by Kenkichi IWASAWA On some modules in the theory of cyclotomic fields by Kenkichi IWASAWA

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Sur la représentation exponentielle dans les corps relativement galoisiens de nombres p-adiques

1939-01-01

Marc Krasner

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Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil I a: Beweise zu Teil I.

1927-01-01

Helmut Hasse

Über die Klassenzahl abelscher Zahlkörper

1985-01-01

Helmut Hasse

Über die Klassenzahl abelscher Zahlkörper

1952-12-31

Helmut Hasse

Nilpotent groups and their generalizations

1940-01-01

Reinhold Baer

Nilpotent finite groups may be defined by a great number of properties. Of these the following three may be mentioned, since they will play an important part in this investigation. … Nilpotent finite groups may be defined by a great number of properties. Of these the following three may be mentioned, since they will play an important part in this investigation. (1) The group is swept out by its ascending central chain (equals its hypercentral). (2) The group is a direct product of p-groups (that is, of its primary components). (3) If S and T are any two subgroups of the group such that T is a subgroup of S and such that there does not exist a subgroup between S and T which is different from both S and T, then T is a normal subgroup of S. These three conditions are equivalent for finite groups; but in general the situation is rather different, since there exists a countable (infinite) group with the following properties: all its elements not equal to 1 are of order a prime number p; it satisfies condition (3); its commutator subgroup is abelian; its central consists of the identity only. A group may be termed soluble, if it may be swept out by an ascending (finite or transfinite) chain of normal subgroups such that the quotient groups of its consecutive terms are abelian groups of finite rank. A group satisfies condition (1) if, and only if, it is soluble and satisfies condition (3) (?2); and a group without elements of infinite order satisfies (1) if, and only if, it is the direct product of soluble p-groups (?3); and these results contain the equivalence of (1), (2) and (3) for finite groups as a trivial special case. If a group without elements of infinite order may be swept out by an ascending chain of subgroups such that each is a normal subgroup of the next one and such that the quotient groups of its consecutive terms are cyclic, then (2) and (3) are equivalent properties, though they no longer imply (1) (?4). If a group satisfies condition (1)-orasuitable weaker conditions-then the elements of finite order in this group generate a subgroup without elements of infinite order which is a direct product of p-groups. A seemingly only slightly stronger condition than (3) is the following property: (3') If S and T are any two subgroups of the group such that T is a subgroup of S and such that there exists at most one subgroup between S and T which is different from both S and T, then T is a normal subgroup of S. Clearly (3') implies (3), though there exist groups which satisfy (3), but not (3'). A closer investigation reveals however that (3') is a much stronger imposition than it seems to be, since it is possible to prove the following theorem: A group, that either does not contain elements of infinite

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l-Extensions of CM-fields and cyclotomic invariants

1980-11-01

Yûji Kida

The Iwasawa Invariant μ p Vanishes for Abelian Number Fields

1979-05-01

Bruce Ferrero , Lawrence C. Washington

Über Automorphismen der Iokal-kompakten abelschen Gruppen

1940-01-01

Makoto Abe

Es soll in der vorliegenden Note die Isomorphie yon dem Automor- phismenring einer lokal-kompakten abelschen Gruppe mit dem ihrer Charaktergruppe gezeigt werden.Es sei G eine lokal-kompakte (additive) abelsche Gruppe, die … Es soll in der vorliegenden Note die Isomorphie yon dem Automor- phismenring einer lokal-kompakten abelschen Gruppe mit dem ihrer Charaktergruppe gezeigt werden.Es sei G eine lokal-kompakte (additive) abelsche Gruppe, die dem zweiten Abzihlbarkeitsaxiom geniigt G* sei die Charaktergrupe

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Zur arithmetischen Theorie der algebraischen Funktionen

1932-12-01

Max Deuring

Lie groups simply isomorphic with no linear group

1936-01-01

Garrett Birkhoff

BIRKHOFF 1. Introduction.One of the most interesting conjectures concerning finite continuous groups is the conjecture that every Lie group is topologically isomorphic with a group of matrices.The proof of this … BIRKHOFF 1. Introduction.One of the most interesting conjectures concerning finite continuous groups is the conjecture that every Lie group is topologically isomorphic with a group of matrices.The proof of this conjecture, even in the small, would establish the truth of the famous conjecture that every Lie group nucleus (or germ) is a piece of a Lie group.J This makes it of interest to know that there exist Lie groups in the large, simply isomorphic even in the purely algebraic sense-and a fortiori topologically isomorphic in the large-with no group of matrices.It is to the proof of this fact that the present note is devoted.2. The Basic Lemma.The proof ultimately rests on the following lemma.LEMMA 1.Let Y be any group of linear transformations.Suppose T contains elements S and T whose commutator R = S" 1 T~1ST is of prime order p, and satisfies SR=RS, TR = RT.Then T is of degree at least p.

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Über nilpotente topologische Gruppen, I

1945-01-01

Kenkiti Iwasawa

Uber nilpotente topolgische Gruppen, I Uber nilpotente topolgische Gruppen, I

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On the Iwasawa Invariants of Totally Real Number Fields

1976-01-01

Ralph Greenberg

Über das verhalten der integrale 1. gattung bei automorphismen des funktionenkörpers

1934-06-01

Catherine Chevalley , André Weil , E. Hecke

Die Einseinheitengruppe von p-Erweiterungen regulärer -adischer Zahlkörper als Galoismodul.

1979-01-01

Kay Wingberg

Article Die Einseinheitengruppe von p-Erweiterungen regulärer -adischer Zahlkörper als Galoismodul. was published on January 1, 1979 in the journal Journal für die reine und angewandte Mathematik (volume 1979, issue 305). Article Die Einseinheitengruppe von p-Erweiterungen regulärer -adischer Zahlkörper als Galoismodul. was published on January 1, 1979 in the journal Journal für die reine und angewandte Mathematik (volume 1979, issue 305).

Topological groups in which multiplication of one side is differentiable

1946-01-01

I. E. Segal

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Axiomatic characterization of fields by the product formula for valuations

1945-01-01

Emil Artin , G. Whaples

Introduction.The theorems of class field theory are known to hold for two kinds of fields: algebraic extensions of the rational field and algebraic extensions of a field of functions of … Introduction.The theorems of class field theory are known to hold for two kinds of fields: algebraic extensions of the rational field and algebraic extensions of a field of functions of one variable over a field of constants.We shall refer to these fields as number fields and function fields, respectively.For class field theory, the function fields must indeed be restricted to those with a Galois field as field of constants; however, we make this restriction only in §5, and until then consider fields with an arbitrary field of constants.In proving these theorems, the product formula for valuations plays an important rôle.This formula states that, for a suitable set of inequivalent valuations | | p, m«i»-i for all numbers a5*0 of the field.For fields of the types mentioned, this product formula is easy to prove.After reviewing this proof ( §1), we shall show ( §2) that, conversely, the number fields and function fields are characterized by their possession of a product formula.Namely, we prove that if a field has a product formula for valuations, and if one of its valuations is of suitable type, then it is either a function field or a number field.This shows that the theorems of class field theory are consequences of two simple axioms concerning the valuations, and suggests the possibility of deriving these theorems directly from our axioms.We do this in the later sections of this paper for the generalized Dirichlet unit theorem, the theorem that the class number is finite, and certain others fundamental to class field theory.This axiomatic method has the decided advantage of uniting the two cases; also, it simplifies the proofs.For example, we avoid the use of either ideal theory or the Minkowski theory of lattice points.Thus these two theories are unnecessary to class field theory, since they are needed only to prove the unit theorem.1. Preliminaries on valuations.If k is any field, then a function | a\, defined for all a£&, is called a valuation of K if

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Sur les groupes topologiques localement compacts

1945-01-01

Jean Braconnier

On p-Adic L-Functions

1969-01-01

Kenkichi Iwasawa

Class Group Relations in Cyclotomic Fields

1952-10-01

Robert E. MacKenzie

On Solvable Extentions of Algebraic Number Fields

1953-11-01

Kenkichi Iwasawa

A Class Number Formula for Cyclotomic Fields

1962-07-01

Kenkichi Iwasawa

Moore-Smith Convergence in General Topology

1937-01-01

Garrett Birkhoff

On the Topological Structure of Solvable Groups

1941-07-01

Claude Chevalley

E. Cartan has shown that a simply connected solvable Lie group is homeomorphic to some Cartesian space.' We want to investigate the structure of a solvable Lie group which is … E. Cartan has shown that a simply connected solvable Lie group is homeomorphic to some Cartesian space.' We want to investigate the structure of a solvable Lie group which is not necessarily simply connected. From a well known theorem,2 it follows that such a group may be considered as the factor group of a solvable simply connected group G by a discrete subgroup D of the center of G. Therefore what we have to do is to look for the possible situation of such a sub-group D in the group G.

On a certainl-adic representation

1973-03-01

Ralph Greenberg

The Theory of Topological Commutative Groups

1934-04-01

L. Pontrjagin

The purpose of the present paper is to make an exhaustive investigation into the structure of continuous, locally compact, commutative groups, satisfying the second axiom of countability.2 1. The purpose of the present paper is to make an exhaustive investigation into the structure of continuous, locally compact, commutative groups, satisfying the second axiom of countability.2 1.

Partially Ordered Abelian Groups

1940-07-01

A. H. Clifford

On Some Invariants of Cyclotomic Fields

1958-07-01

Kenkichi Iwasawa

On solvable extensions of algebraic number fields

1953-01-01

Ichirō Satake , Genjiro Fujisaki , Kazuya Katô +2 more

A note on class numbers of algebraic number fields

1956-03-01

Kenkichi Iwasawa

Lectures on P-Adic L-Functions. (AM-74)

1972-12-31

Kinkichi Iwasawa

An especially timely work, the book is an introduction to the theory of p-adic L-functions originated by Kubota and Leopoldt in 1964 as p-adic analogues of the classical L-functions of … An especially timely work, the book is an introduction to the theory of p-adic L-functions originated by Kubota and Leopoldt in 1964 as p-adic analogues of the classical L-functions of Dirichlet. Professor Iwasawa reviews the classical results on Dirichlet's L-functions and sketches a proof for some of them. Next he defines generalized Bernoulli numbers and discusses some of their fundamental properties. Continuing, he defines p-adic L-functions, proves their existence and uniqueness, and treats p-adic logarithms and p-adic regulators. He proves a formula of Leopoldt for the values of p-adic L-functions at s=1. The formula was announced in 1964, but a proof has never before been published. Finally, he discusses some applications, especially the strong relationship with cyclotomic fields.

Cohomologie Galoisienne non Commutative

1965-01-01

Jean-Pierre Serre

A note on capitulation problem for number fields

1989-01-01

Kenkichi Iwasawa

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Über die Automorphismen eines algebraischen Funktionenkörpers von Primzahlcharakteristik.

1938-01-01

Hermann Schmid

Article Über die Automorphismen eines algebraischen Funktionenkörpers von Primzahlcharakteristik. was published on January 1, 1938 in the journal Journal für die reine und angewandte Mathematik (volume 1938, issue 179). Article Über die Automorphismen eines algebraischen Funktionenkörpers von Primzahlcharakteristik. was published on January 1, 1938 in the journal Journal für die reine und angewandte Mathematik (volume 1938, issue 179).