L. C. Washington. Introduction to Cyclotomic Fields. SpringerVerlag, Grad. Texts Math. 83, Second Edition, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... one of the Bernoulli numbers B 2 , B 4 , B p 1 (2) if we write the characteristic series of A for the cyclotomic Z p extension in the form f = T cp)u, where u is a unit power series and # is chosen to satisfy # = # 1 p for all p power roots of unity #, then c 1 (mod p) See [W] Theorem 10.16 and Theorem 8.25. The computations in [BCEMS] verify Vandiver s conjecture for all primes less than 12,000,000, and condition (1) is satisfied by about 30 of those primes (about 61 of them are regular, in which case A = 0) Iwasawa and Sims [IS] tabulate the congruence classes ....

....have already seen that K N# , this proves the lemma. To prove the claim, we show that K# contains H, the p Hilbert class field of K. It follows from well known results in the theory of cyclotomic fields that H is generated by the p th th root of a cyclotomic unit. The reflection principle, [W], Theorem 10.9, implies that p does not divide h . The claim now follows from Theorems 10.16, 8.2, and 8.25. Since the p th roots of all the cyclotomic units in K are contained in K# , this implies that H K# . 6. The Main Theorem. Theorem 26. Let K# K be a multiple Z p extension of K, ....

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L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York, 1982. Department of Mathematics, University of Arizona, Tucson, AZ, 85721 21

....for a classi cation of symmetries of quasilattices has been pointed out by Niizeki [11] Let an LI class of tilings in 2D be given with n fold (generalized) rotation symmetry. In the minimal rank case, the corresponding LTM is (up to a similarity transformation) the ring of cyclotomic integers [12] Z[ fa 1 a 2 a 3 a (n) j = e 2 i=n ; a k 2 Z; k = 1; n)g; where denotes Euler s totient function from number theory. Regarding Z[ as projection image of a periodic lattice of higher dimension, it is natural to identify the generalized symmetries ....

....look for the number theoretic counterpart of the torus equation (2) in the cut and project scheme. It translates to a linear equation in the cyclotomic eld Q( The determinant (3) equals the absolute algebraic norm in the corresponding ring of cyclotomic integers, which we now de ne (see e.g. [12, 13]) The ring Z[ the integer polynomials in ) consists of all algebraic integers of the cyclotomic eld Q ( the rational functions in . Now, is a root of the n th cyclotomic polynomial, P n (x) which is integral, irreducible over Z and of degree (n) The other solutions are the (n) 1 ....

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L. C. Washington, \Introduction to Cyclotomic Fields", Springer, New York, 2nd edition (1997).

....ideals of Z[i] of norm m. The answer is given by the Dedekind zeta function of the quadratic (or cyclotomic) field Q (i) i.e. one obtains [3, Eq. 7) 1 2 (4) 20 The corresponding question for the module Z[x ] is answered by the Dedekind zeta function [18] of the cyclotomic field Q(x ) which reads [3, Eq. 10) 5) 4 These two generating functions also have an interpretation in terms of colourings: a is (up to permutation) the number of colourings of the square lattice where one colour occupies a ....

....lattice where one colour occupies a sublattice of Z of index m and the remaining colours occupy its cosets. The function (m) in turn, counts the colourings of Z[x ] via submodules of index m, all of which are similarity submodules because Z[x ] as Z[i] is a principal ideal domain, see [18, 7] for details. For a discussion of the corresponding colour groups, see [12] This approach can be extended to all cyclotomic fields with class number one (the corresponding rings of integers are then principal ideal domains) see [18, 3, 5] and references given there, for details. 4. Counting ....

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L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Springer, New York (1997).

....degree 2) Sublattices now correspond to subgroups of nite index, and those invariant under fourfold rotation (i.e. under multiplication by i) correspond to ideals. They are the subgroups a Z[i] with a a for all 2 Z[i] for background material on the concepts and results used we refer to [7, 5, 18]. Consequently, counting all square sublattices of Z of index m is the same as counting all ideals of the ring Z[i] of nite index m, where m = norm(a) is the (number theoretic) norm of a (which equals the number of residue classes of a in Z[i] As the ring Z[i] is a principal ideal domain ....

....generality: other planar cases Having desribed the square lattice in detail, we shall now generalize our approach to other Z modules of the plane, namely those with n fold rotational symmetry, n 2. Though many things are similar here, the number theoretic background is a lot more involved, see [18]. Let us nevertheless consider the Z span of a regular n star, M n : Z 1 Z : Z n 1 ; 8) where = exp(2 i=n) This is called the standard n fold symmetric module of the plane [11] It is a Z module of rank (n) where denotes Euler s totient function [5] and it is ....

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L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Springer, New York (1997).

....theory of Z p extensions. In order to keep the paper in a reasonable length, we will just provide background material on both areas which is necessary for later discussions. For further detail and any unexplained terminology, we refer the reader to O Meara s book [OM] and Washington s book [Wa]. We also refer the reader to Iwasawa s original papers [I1 4] for more information on Z p extensions. From now on, F is always either a number field or the completion of a number field at one of its prime spot. In the later case, we will simply say that F is a local field. The ring of integers ....

....) Theta Ln and hence Sigma n Sigma n 1 . Let Sigma 1 be the union of the Sigma n s. It is not hard to see that Sigma 1 is a Galois extension of F . Therefore, Gal( Sigma 1=F1 ) is a module. As Sigma 1=F1 is unramified outside T , Gal( Sigma 1=F1 ) is a finitely generated module (see [Wa] or [I4] Let us first assume that L has Type II reduction at all 2 D . We claim that Sigma n is the maximal elementary 2 extension of F n inside Sigma 1 . Let M n be the maximal elementary 2 extension of F n inside Sigma 1 . Then Sigma n M n . If we let V n be the open subgroup in J ....

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L. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, Springer Verlag, New York, 1982.

....A achieves maximum coding gain when using QAM constellations: BA For , Construction A can not achieve the upper bound (9) of coding gains over QAM constellations. Proof: When by Lemma 4. When # s , not all roots of have modulus 1 as by Lemma 1. 6 in [8]. Hence, we have w K by Lemma 5. Therefore, the upper bound (9) can not be achieved by (12) in Theorem 3. Next, we state without proof a theorem which provides lower bounds on coding gains of Construction B. 3 s N o 2 2 o 2 o K 2 2 s U o 2 ....

L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Math.83, Second edition, Springer-Verlag, 1997.

....the Bernoulli numbers B 2 ; B 4 ; B p Gamma1 (2) the characteristic power series of A for the cyclotomic Z p extension has the form f = T Gamma cp)u, where u is a unit power series and c 6j 1 (mod p) and fl is chosen to satisfy i fl = i 1 p for all p power roots of unity i. See [W] Theorem 10.16 and Theorem 8.25. The computations in [BCEMS] verify Vandiver s conjecture for all primes less than 12,000,000, and condition (1) is satisfied by about 30 of those primes (about 61 of them are regular, in which case A = 0) Iwasawa and Sims [IS] tabulate the congruence classes ....

....seen that K ae N1 , this proves the lemma. 19 To prove the claim, we show that K1 contains H, the p Hilbert class field of K. It follows from well known results in the theory of cyclotomic fields that H is generated by the p th th root of a cyclotomic unit. The reflection principle, [W], Theorem 10.9, implies that p does not divide h . The claim now follows from Theorems 10.16, 8.2, and 8.25. Since the p th roots of all the cyclotomic units in K are contained in K1 , this implies that H ae K1 . 6. The Main Theorem. Theorem 26. Let K1=K be a multiple Z p extension of K, ....

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L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York, 1982. Department of Mathematics, University of Arizona, Tucson, AZ, 85721 21

....of the cyclotomic Z p extension of Q ( p ) for primes up to 4 million. In fact, for these primes, p) is equal to the index of irregularity of the prime p (see Section 2.1.1) In particular, there seems to be no shortage of cyclotomic elds with invariant 1. Using the tables in the back of [28] we see that 165 of the rst 219 irregular primes have (p) 1, and in general this should be the case for approximately 30 of all primes. Remark 3. The assumptions of Vandiver s conjecture and (p) 1 imply that A(K) Z=p m Z for some m (see Proposition 2.2.1) and hence there is only ....

....16 Chapter 2 BACKGROUND In this chapter we review many of the facts and constructions needed for a full understanding of both the statement of Greenberg s conjecture as well as the proof in Chapter 3 of the main result. The main reference for Sections 2.1 and 2. 2 are the text of Washington [28], as well as that of Lang [18] For more details on the content of Section 2.4 see [21] The content of Section 2.3 seems to be folklore, although I found no single source containing the derivations given here (Section 13.1 of [28] was helpful for the details of Section 2.3.2) Some needed ....

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L. Washington. Introduction to Cyclotomic Fields, Second Edition, Graduate Texts in Mathematics, Springer-Verlag, New York, 1997.

....Conjecture was proved independently and with di#erent methods by Deligne [18, V] and Hayes [11] In the case of number fields, this conjecture is far from being proved. The statement has been known to hold true for a long time if k = Q, as a result of the classical theorem of Stickelberger (see [20]) Wiles developed a series of results and techniques in [21] which lead to a proof of the conjecture above in the case where K is a CM field, k is totally real, # # G , and S 0 satisfies extra hypotheses. Finally, by using the techniques developed in [21] Greither proves the conjecture ....

L. Washington, Introduction to cyclotomic fields, second edition, GTM 83, Springer-Verlag, New York, Berlin, Heidelberg.

....the assertion. b) We put m # = 4 in Theorem 3.3 and get the assertion. c) Put m # = q b in Theorem 3.3. Then u = #(p a ) w 0 = 2q c and e = q 1)q c 1 , since o q b (p) q b c . # Example 3. 5 We choose an example which can be compared with the table of relative class numbers in [21]. By Corollary 3.4 c) we have h 2311 # 0 (mod 2 4 11 4 ) The table shows that 2 4 , 11 4 are actually the highest powers of 2 respectively 11 dividing h 2311 . Another approach to class number factors can be found in [4, 6] the method is to use Abhyankar s lemma to construct ....

L.C. Washington, "Introduction to Cyclotomic Fields," Graduate Texts in Math. No. 83. Springer Verlag, Berlin/New York/Heidelberg, 1997. 18

.... 0 p ( is [P 0 ] torsion, see x 5 and x 6 of Chapter 5 in [La] As a consequence of the famous Ferrero Washington theorem in [FW] Gamma saying that the Iwasawa invariant p vanishes for Abelian number fields Gamma and the structure theorem for Iwasawa modules Gamma e.g. Theorem 13.12 of [Wa] Gamma one obtains that X 0 p ( p has finite order. A characterization of the Fitting ideal of X 0 p ( is given in Lemma 3.7 of [Gri] Lemma 4.6. Greither) Let M be a [P 0 ] torsion module of projective dimension less or equal to 1. Suppose that M=p is finite and that OE 2 [P 0 ] ....

....write Gamma1 (p) for Gamma1 (F rob p ) where F rob p is the Frobenius element of G associated to p. Proof. Write = 0 ae where 0 is a character of Gal(E 0 =Q) and ae is a character of Gal(E=E 0 ) By the characterization of F Gamma1 n 2 1 and Theorems 4.2 and 5. 11 in [Wa] we find: Gamma1 (ff ) G 0 Gamma1 n 2 1( fl) n 2 1 ae(fl) Gamma 1) L p ( Gamma n 2 ; Gamma1 n 2 1 ) 1 Gamma p n 2 Gamma1 (p) L( Gamma n 2 ; Gamma1 ) Gamma(1 Gamma p n 2 Gamma1 (p) B n 2 1; Gamma1 n 2 1 : On ....

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L. C. Washington, Introduction to Cyclotomic Fields, 2nd Ed., Graduate Texts in Mathematics, vol. 83, Springer Verlag, 1997.

....method ( We] HW] Appearing in manuscripts of Jayadeva and Bh askara dating back to the 11th and 12th centuries (and rediscovered independently much later by Fermat) it is one of the great contributions of Indian mathematics and civilization ( We] I.IX) 2. The circular unit method ( Ma] [Was]) Suppose for simplicity that D 1 (mod 4) is square free, and let D (n) n D be the quadratic Dirichlet character. The following theorem of Gauss, intimately connected with quadratic reciprocity, is the basis for the circular unit method. Theorem 1.1 Every quadratic eld is contained ....

L. Washington, Introduction to Cyclotomic Fields, GTM 83, Springer-Verlag, 1982.

....1 l. But is not isometric to , for the determinant of is Gamma a 2 ; ff Delta for some ff, and a 2 is non zero in the ideal class group since r is odd. Hence k 6= l. A similar, but more involved, argument shows that k r 6= l r . Many examples may be obtained from the tables in [46] or [47]. 7. Spinning and Branched Cyclic Covers First we recall a definition of spinning. Let k be the n knot Gamma S n 2 ; S n Delta and let B be a regular neighbourhood of a point on S n such that (B; B S n ) is an unknotted ball pair. Then the closure of the complement of B in S n 2 ....

L.C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York-Heidelberg-Berlin, 1982.

....we drew from this hypothesis in the n = 0 case are still true. 5 That does it; now we see how to study all of the X n s by studying the # module X. We will now discuss the structure theory of # modules. First we discuss # itself. All of the following results are nicely explained and proved in [W] 7.1 and 13.2. Recall that # # = lim # Z p [# # n ] It is clear that Z p [# # n ] # = Z p [T ] T 1) p n 1) The isomorphism is induced by # 0 # n ## T 1. The fact that (T 1) p n 1 divides (T 1) p j 1 for j i defines maps from Z p [T ] T 1) p j 1) to ....

L. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics 83 (

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L. C. Washington. Introduction to Cyclotomic Fields. SpringerVerlag, Grad. Texts Math. 83, Second Edition, 1997.

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L. C. Washington. Introduction to Cyclotomic Fields. Springer-Verlag, Grad. Texts Math. 83, Second Edition, 1997. MR 97h:11130.

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L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, NY, 1982.

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L. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics

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L. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York, 1982.

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L. Washington. Introduction to Cyclotomic Fields. Second edition, volume 83 of Graduate Texts in Math. Springer-Verlag: New York, 1997.

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L. C. Washington, "Introduction to Cyclotomic Fields," in Graduate Texts in Math, 2nd ed. New York: Springer-Verlag, 1997, vol. 83.

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L. C. Washington. Introduction to cyclotomic fields. Graduate Texts in Mathematics 83, Springer-Verlag, Berlin/New York 1980.

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L. C. Washington, Introduction to Cyclotomic Fields, Spinger-Verlag, Graduate Texts in Mathematics 83, 1982.

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L. C. Washington, Introduction to Cyclotomic Fields , 2nd edition, Springer, 1997.

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L. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, 1997.

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