Maximal Real Subfield Research Articles

BERNOULLI-GOSS POLYNOMIAL AND CLASS NUMBER OF CYCLOTOMIC FUNCTION FIELDS

Let k = Fq(T),q=pn, and let K=k((?)p)be the cyclotomic function field with conduc-tor P = P(T), and suppose K+ is the maximal real subfield of K, hp(h+p) is the class number of divisor group (of degree zero) of K(K+), and h-p=hp/h+p(∈ Ⅱ). This paper proves that for any fixed q≥3, there exist infinite many irreducible manic polynomial P∈Fq[T] such that p\h+p and pq-2\h-p. In addition, all regular quadratic irreducible polynomials in Fq[T] for 2≤p≤269 are determined.

Iwasawa invariants of CM fields

Let K be a CM field with K + its maximal real subfield. Let λ, λ + be the Iwasawa λ-invariants for the cyclotomic Z p -extension of K, K +, respectively. We set λ − = λ − λ +. We show that under certain conditions, λ + ≤ λ − − 1. We use this inequality to give criteria which imply that λ + = 0.

Note on the class-number of the maximal real subfield of a cyclotomic field, II

For an integer m > 2, we denote by C(m) and H(m) the ideal class group and the class-number of the fieldK = Q(ζm + ζm−1)respectively, where ζm is a primitive m-th root of unity. Let q be a prime and /Q be a real cyclic extension of degree q. Let C() and h() be the ideal class group and the class-number of . In this paper, we give a relation between C() (resp. h()) and C(m) (resp. H(m)) in the case that m is the conductor of (Main Theorem). As applications of this main theorem, we give the following three propositions. In the previous paper [4], we showed that there exist infinitely many square-free integers m satisfying n|H(m) for any given natural number n. Using the result of Nakahara [2], we give first an effective sufficient condition for an integer m to satisfy n|H(m) for any given natural number n (Proposition 1). Using the result of Nakano [3], we show next that there exist infinitely many positive square-free integers m such that the ideal class group C(m) has a subgroup which is isomorphic to (Z/nZ)2 for any given natural number n (Proposition 2). In paper [4], we gave some sufficient conditions for an integer m to satisfy 3|H(m) and m≡l (mod 4). In this paper, using the result of Uchida [5], we give moreover a sufficient condition for an integer m to satisfy 4|H(m) and m ≡ 3 (mod 4) (Proposition 3). Finally, we give numerical examples of some square-free integers m satisfying 4|H(m) and m ≡ 3 (mod 4).

A remark on the class-number of the maximal real subfield of a cyclotomic field

A remark on the class-number of the maximal real subfield of a cyclotomic field

On the class number of some quaternion group rings

the generalized quaternion group of order 4p. The aim of this paper is to calculate the class number of the integral group ring ZQ,; by this we mean the order h(ZQ,) of C/(ZQ,), the group of stable isomorphism classes of locally free ZQ,-modules. To state the result, we introduce the following notations: [ is a primitive pth root of unity, Z, = Z[c], Q,= a(l), 0; = Q, n R = maximal real subfield of Q,, Zp’ = Z, n 0:) Y = (1 5) = unique prime ideal over p in H,, Y + = Y n Zt, CI’ (Zp+ ) = narrow ideal class group of Z; = Z,+-ideals in QT modulo principal ideals with a totally positive generator, h + (77: ) = order of Cl + (Z,’ ). It is well known that there is an exact sequence

Quintic polynomials and real cyclotomic fields with large class numbers

We study a family of quintic polynomials discoverd by Emma Lehmer. We show that the roots are fundamental units for the corresponding quintic fields. These fields have large class numbers and several examples are calculated. As a consequence, we show that for the prime p = 641491 p = 641491 the class number of the maximal real subfield of the pth cyclotomic field is divisible by the prime 1566401. In an appendix we give a characterization of the "simplest" quadratic, cubic and quartic fields.

Note on the class-number of the maximal real subfield of a cyclotomic field

Note on the class-number of the maximal real subfield of a cyclotomic field

Non monogénéité de l'anneau des entiers des extensions cycliques de [formula omitted] de degré premier l ≥ 5

Let K/ Q be a cyclic extension of degree l. Let Z K be the ring of integers of K. We say that Z K has a power basis (or is monogenic) if there exists θ ∈ Z K such that Z K = Z [θ]. We show that if l ≥ 5 is a prime, then Z K has no power basis, except in the well-known case where K is the maximal real subfield of a cyclotomic field; that is to say, if l is given, there exists, at most, one field K such that Z K has a power basis: the field Q 0 (2l + 1) , when 2 l + 1 is prime (e.g., for l = 5, Z K has a power basis only for the field K = Q 0 (11) , and for l = 7, Z K never has a power basis).

A Lower Bound for the Class Number of Certain Cubic Number Fields

Let K be a cyclic number field with generating polynomial X3 a 3 32 - a + 3 X - 1 2 2 and conductor m. We will derive a lower bound for the class number of these fields and list all such fields with prime conductor m = (a2 + 27)/4 or m = (1 + 27b2)/4 and small class number. 1. Introduction. Let h.,, denote the class number of the cyclotomic field Q(,m), and h +, the class number of its maximal real subfield Q(cos(2T/m)). It is a well-known conjecture of Vandiver that p + h+ holds for all primes p E P. This is a customary assumption for proving the second case of Fermat's Last Theorem (for more details see Washington (16)). Since h+ grows slowly (h+ = 1 for p 1 the exact value of h + is known without using GRH. Masley suggested that P P perhaps h + 1. Using the quadratic subfield, Ankeny, Chowla and Hasse (1) showed that h + > 1 if p belongs to certain quadratic sequences in N, and S.-D. Lang (9) and Takeuchi (15) found more such sequences. Similar results were

Asymptotic Results for Class Number Divisibility in Cyclotomic Fields

AbstractLet n ≥3 and m≥3 be integers. Let Kn be the cyclotomic field obtained by adjoining a primitive nth root of unity to the field of rational numbers. Let denote the maximal real subfield of Kn. Let hn (resp., ) denote the class number of Kn (resp., ). For fixed m we show that m divides hn and hn for asymptotically almost all n. Also for those Kn and with a given number of ramified primes, we obtain lower bounds for certain types of densities for m dividing hn and .

Exponential growth of the 𝑙-rank of the class group of the maximal real subfield of cyclotomic fields

Exponential growth of the 𝑙-rank of the class group of the maximal real subfield of cyclotomic fields

On the first factor of the class number of a cyclotomic field

Let p p be an odd prime. h 1 ( p ) {h_1}(p) is the first factor of the class number of field Q ( ζ p ) Q({\zeta _p}) . We proved that \[ h 1 ( p ) ⩽ { 2 p ( p − 1 8 ( 2 l / 2 + 1 ) 4 / l ) ( p − 1 ) / 4 , if l is even, 2 p ( p − 1 8 ( 2 l − 1 ) 2 / l ) ( p − 1 ) / 4 , if l is odd . {h_1}(p) \leqslant \left \{ \begin {gathered} 2p{\left ( {\frac {{p - 1}} {{8{{({2^{l/2}} + 1)}^{4/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is even,}} \hfill \\ 2p{\left ( {\frac {{p - 1}} {{8{{({2^l} - 1)}^{2/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is odd}}{\text {.}} \hfill \\ \end {gathered} \right . \] From that we obtain h 1 ( p ) ⩽ 2 p ( ( p − 1 ) / 31.997158 … ) ( p − 1 ) / 4 {h_1}(p) \leqslant 2p{((p - 1)/31.997158 \ldots )^{(p - 1)/4}} which is better than Carlitz’s and Metsänkyla’s results. For the fields Q ( ζ 2 n ) Q({\zeta _{{2^n}}}) and Q ( ζ p n ) ( n ⩾ 2 ) Q({\zeta _{{p^n}}})(n \geqslant 2) , we get the similar results.

Induced Quaternion Algebras in the Schur Group

Let K be a finite, imaginary and abelian extension of the rational number field Q, and let M be the maximal real subfield of K. It is well known that each element of order 2 in S(K), the Schur group of K, is induced from an element of order 2 in B(M), the Brauer group of M; i.e., if D is a quaternion division algebra central over K such that its class [D] in B(K) is in fact in S(K) then [D] = [B ⊗MK] where B is a quaternion division algebra with [B] ∈ B(M). A natural question to ask is: “When is every element of S(K) of order 2 induced from S(M)?” The main result of this paper is to provide necessary and sufficient conditions for this to occur when G(L/K), the Galois group of L over K, is cyclic where L is the smallest root of unity field containing K.

The class number of cyclotomic function fields

Let k be a rational function field over a finite field. Carlitz and Hayes have described a family of extensions of k which are analogous to the collection of cyclotomic extensions { Q( ζ m )| m ≥ 2} of the rational field Q. We investigate arithmetic properties of these “cyclotomic function fields.” We introduce the notion of the maximal real subfield of the cyclotomic function field and develop class number formulas for both the cyclotomic function field and its maximal real subfield. Our principal result is the analogue of a classical theorem of Kummer which for a prime p and positive integer n relates the class number of Q( ζ p n + ζ p n −1), the maximal real subfield of Q( ζ p n ), to the index of the group of cyclotomic units in the full unit group of Z[ ζ p n ].

On the Class-Number of the Maximal Real Subfield of a Cyclotomic Field

Let p be an integer and let H(p) be the class-number of the fieldwhere ζp is a primitive p-th root of unity and Q is the field of rational numbers. It has been proved in [1] that if p = (2qn)2 + 1 is a prime, where q is a prime and n > 1 an integer, then H(p) > 1. Later, S. D. Lang [2] proved the same result for the prime number p = ((2n + 1)q)2 + 4, where q is an odd prime and n ≧ 1 an integer. Both results have been obtained in the case p ≡ 1 (mod 4).In this paper we shall prove the similar results for a certain prime number p ≡ 3 (mod 4).We designate by h(p) the class-number of the real quadratic field

Grössencharaktere und Relativklassenzahl abelscher Zahlkörper

Let K be an absolute abelian number field and K 0 its maximal real subfield with respective class numbers h and h 0 and G the group of all algebraic Hecke charakters of K. By means of Gaussian sums a subgroup of G is constructed whose index in G describes the relative class number h ∗ = h h 0 of K. This contains a generalization of a formula of Iwasawa for cyclotomic fields with prime power conductor. In a certain sense the formula obtained is an analogue of Leopoldt's description of h 0 by circular units.

Computing the number of totally positive circular units which are squares

An efficient method for computing the number of invariants of the quotient group C C ⌢ T , where C is the group of circular units in the maximal real subfield of the qth cyclotomic field, q is a prime, and T is the group of totally positive units, is developed by establishing an isomorphism between C C ⌢ T and a specific ideal in the algebra F[x] 〈x p + 1〉 , where F is the Galois field of two elements and p = (q − 1) 2 . Based on computations using this method, it is conjectured that every totally positive circular unit is a square if p is a prime.

Note on the class-number of the maximal real subfield of a cyclomatic field.

Article Note on the class-number of the maximal real subfield of a cyclomatic field. was published on January 1, 1977 in the journal Journal für die reine und angewandte Mathematik (volume 1977, issue 290).

On the integral basis of the maximal real subfield of a cyclotomic field.

Article On the integral basis of the maximal real subfield of a cyclotomic field. was published on January 1, 1976 in the journal Journal für die reine und angewandte Mathematik (volume 1976, issue 286-287).

Units of modulus 1

If k is an algebraic number field which is normal over the field of rational numbers then it is shown that k has nontrivial units of modulus 1 if and only if the maximal real subfield of k is also a normal extension of the rationals. A characterization of the units is given for fields which satisfy the above conditions. A new proof of Kummer's Theorem on the units of cyclotomic fields is also obtained.