a parametric family of quintic thue equations for an integral parameter t z we investigate the family of thue equations f x y x 5 t 1 2 x 4 y 2t 3 4t 4 x 3 y 2 t 4 t 3 2t 2 4t 3 x 2 y 3 t 3 t 2 5t 3 xy 4 y 5 1 originating from emma lehmer s family of quintic fields and show that for t 3 28 10 15 the only solutions are the trivial ones with x 0 or y 0 our arguments contain some new ideas in comparison with the standard methods for thue families which gives this family a special interest

note on a polynomial of emma lehmer recently emma lehmer constructed a parametric family of units in real quintic fields of prime conductor p t 5t i5t 25 25 as translates of gaussian periods later schoof and washington showed that these units were fundamental units in this note we observe that lehmer s family comes from the covering of modular curves x 25 y0 25 this gives a conceptual explanation for the existence of lehmer s units they are modular units which have been studied extensively by relating lehmer s construction with ours one finds expressions for certain gauss sums as values of modular units on y 25 1 lehmer s polynomial throughout the discussion we fix a choice of primitive th roots of

class numbers of real cyclotomic fields of prime conductor the class numbers hl of the real cyclotomic fields q l l 1 are notoriously hard to compute indeed the number hl is not known for a single prime l 71 in this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields q l l 1 for the primes l 10 000 it is quite likely that these subgroups are in fact equal to the class groups themselves but there is at present no hope of proving this rigorously in the last section of the paper we argue on the basis of the cohen lenstra heuristics that the probability that our table is actually a table of class numbers hl is at least 98

computation of class numbers of quadratic number fields we explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields we therefore end up with a simpler and faster method for computing class numbers of quadratic number fields we also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields

connection between gaussian periods and cyclic units this paper finds that all known parametric families of units in real quadratic cubic quartic and sextic fields with prime conductor are linear combinations of gaussian periods and exhibits these combinations this approach is used to find new units in the real quintic field for prime conductors p n4 5n3 15n2 25n 25

witt groups of smooth projective quadrics abstract let k be a commutative ring containing 1 2 in this paper we construct exact sequences connecting witt groups of smooth projective quadrics over k and clifford algebras the exact sequences have an application to a classical problem in quadratic form theory the witt kernel of function fields of quadrics we also apply the exact sequences to compute witt groups of certain kinds of smooth projective quadrics

jacobi sums and new families of irreducible polynomials of gaussian periods let m 2 m an m th primitive root of 1 q 1 mod 2m a prime number s sq a primitive root modulo q and f fq q 1 m we study the jacobi sums j a b q 1 k 2 a inds k b inds 1 k m 0 a b m 1 where ind s k is the least nonnegative integer such that s ind s k k mod q we exhibit a set of properties that characterize these sums some congruences they satisfy and a maple program to calculate them then we use those results to show how one can construct families p q x q e p of irreducible polynomials of gaussian periods i f 1 j 0 si mj q of degree m where p is a suitable set of primes 1 mod 2m we exhibit examples of such families for several small values of m and give a maple program to construct more of them

a family of cyclic quartic fields arising from modular curves we study a family of cyclic quartic fields arising from the covering of modular curves x1 16 xo 16 an integral basis and a fundamental system of units are found it is shown that a root of the quartic polynomial we construct is a translate of a cyclotomic period by an integer of the quadratic subfield of the quartic field recently 0 lecacheux 9 10 and h darmon 4 showed how to use coverings of modular curves to obtain cyclic extensions of q in particular they were able to give a geometric construction of a family of cyclic quintic fields discovered by e lehmer 11 the covering x1 n x0 n for n 2 has degree q n 2 and group z nz x i 1 for the quintic case they took n 25 which gave a cyclic covering of degree 10 then took the subcovering of degree 5 an important ingredient in the construction was the fact that xo 25 has genus 0 this also occurs for n 1 10 12 13 16 18 these all give trivial or quadratic coverings except for n 7 9 13 16 18 the values n 7 9 18 yield cubic extensions and can be shown to yield the family of polynomials x3 ax2 a 3 x 1 namely the simplest cubic fields 17 however it should be remarked that every cyclic cubic extension of q comes from a polynomial of this form if a is allowed to be rational similarly we are guaranteed that the quadratic extensions obtained from the coverings mentioned above correspond to polynomials of the form x2 ax 1 with a rational the case n 13 is treated by lecacheux 9 it might be suspected that the sextic fields she obtains are the same as the simplest sextics constructed by m n gras 6 however these latter fields were found by taking the fixed field of an element of order 6 in pgl2 q aut q x therefore they come from a covering of curves of genus 0 but x1 13 has genus 2 alternatively these sextic fields must be different because the discriminants of the quadratic and cubic subfields are different in the present paper we study the case n 16 as above it might be hoped that this case would give a geometric construction of the quartic fields studied received march 14 1990 revised october 22 1990 1980 mathematics subject classification 1985 revision primary 1 i r 16 research supported in part by n s f 1991 american mathematical society 0025 5718 91 1 00 25 per page

families of cyclic polynomials obtained from geometric generalization of gaussian period relations a general method of constructing families of cyclic polynomials over q with more than one parameter will be discussed which may be called a geometric generalization of the gaussian period relations using this we obtain explicit multi parametric families of cyclic polynomials over q of degree 3 e 7 we also give a simple family of cyclic polynomials with one parameter in each case by specializing our parameters

sampling properties of directed networks for many real world networks only a small sampled version of the original network may be investigated those results are then used to draw conclusions about the actual system variants of breadth first search bfs sampling which are based on epidemic processes are widely used although it is well established that bfs sampling fails in most cases to capture the in component s of directed networks a description of the effects of bfs sampling on other topological properties is all but absent from the literature to systematically study the effects of sampling biases on directed networks we compare bfs sampling to random sampling on complete large scale directed networks we present new results and a thorough analysis of the topological properties of seven complete directed networks prior to sampling including three versions of wikipedia three different sources of sampled world wide web data and an internet based social network we detail the differences that sampling method and coverage can make to the structural properties of sampled versions of these seven networks most notably we find that sampling method and coverage affect both the bow tie structure and the number and structure of strongly connected components in sampled networks in addition at a low sampling coverage i e less than 40 the values of average degree variance of out degree degree autocorrelation and link reciprocity are overestimated by 30 or more in bfs sampled networks and only attain values within 10 of the corresponding values in the complete networks when sampling coverage is in excess of 65 these results may cause us to rethink what we know about the structure function and evolution of real world directed networks

pairings in triangular witt theory abstract given a product k l m between triangulated categories with duality we show that under some conditions there exist naturally two different pairings w i k w j l w i j m where w denotes the triangulated witt functor of balmer p balmer k theory 19 2000 311 363 our main example of such a situation is the case that k l m is the bounded derived category of vector bundles over a scheme x and is the derived tensor product the derived witt groups of this scheme w x i z w i x become a graded skew commutative ring with two different product structures which are both equally natural in the last section we prove then a projection formula for our product and show as an application that a springer type theorem is true for the derived witt groups too

efficient computation of root numbers and class numbers of parametrized families of real abelian number fields let k m be a parametrized family of simplest real cyclic cubic quartic quintic or sextic number fields of known regulators e g the so called simplest cubic and quartic fields associated with the polynomials p m x x3 mx2 m 3 x 1 and p m x x 4 mx 3 6x 2 mx 1 we give explicit formulas for powers of the gaussian sums attached to the characters associated with these simplest number fields we deduce a method for computing the exact values of these gaussian sums these values are then used to efficiently compute class numbers of simplest fields finally such class number computations yield many examples of real cyclotomic fields q p of prime conductors p 3 and class numbers h p greater than or equal to p however in accordance with vandiver s conjecture we found no example of p for which p divides h p

the mayer vietoris principle for grothendieck witt groups of schemes we prove localization and zariski mayer vietoris for higher gro thendieck witt groups alias hermitian k groups of schemes admitting an ample family of line bundles no assumption on the characteristic is needed and our schemes can be singular along the way we prove additivity fibration and approximation theorems for the hermitian k theory of exact categories with weak equivalences and duality

cyclotomy and delta units in this paper we examine cyclic cubic quartic and quintic number fields of prime conductor p containing units that bear a special relationship to the classical gaussian periods qj qj 1 c is a unit for periods q and c e z

a family of real 2ⁿ tic fields we study the family of polynomials pn x a 9 x i 2n a 3 x i 2n and determine when p x a a e z is irreducible the roots are all real and are permuted cyclically by a linear fractional transformation defined over the real subfield of the 2nth cyclotomic field the families of fields we obtain are natural extensions of those studied by m n gras and y y shen but in general the present fields are non galois for n 4 from the roots we obtain a set of independent units for the galois closure that generate an almost fundamental piece of the full group of units finally we discuss the two examples where our fields are galois namely a 2n and a 24 239

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 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

class number computations of real abelian number fields in this paper we describe the calculation of the class numbers of most real abelian number fields of conductor 200 the technique is due to j m masley and makes use of discriminant bounds of a m odlyzko in several cases we have to assume the generalized riemann hypothesis introduction it is well known that the class number h of an abelian number field can be written as h h h where h is the class number of the maximal real subfield k of k and h is an integer we can determine the relative class number h in a straightforward way using the complex analytic class number formula see 7 kap iii or 9 chapter 3 section 3 for the full cyclotomic fields q n with 4o n 256 and their subfields one can deduce hfrom the tables of g schrutka von rechtenstamm 15 here dn denotes a primitive nth root of unity and 40 is the euler function for the class number factor h the complex analytic class number formula is less useful since it requires that the units of k be known alternative techniques have been developed by j m masley 13 who computed the class number of almost all real cyclic number fields of conductor 100 here the conductor of k is the least f for which k c q tf in this paper we apply masley s techniques with a few additions to determine the class numbers of a large collection of real abelian fields of conductor 200 see section 1 for a precise statement of our results some of which assume the generalized riemann hypothesis an important ingredient of masley s method is the use of discriminant lower bounds proved by a m odlyzko 14 these lead to an upper bound for the class number of a real abelian number field provided that its conductor or more precisely its root discriminant see 13 section 1 is sufficiently small it follows that this method can only be used for a finite number of real abelian number fields the existence of infinite class field towers shows that this remains true after any future improvement of odlyzko s bounds in fact examples of j martinet 12 show that the method will never apply to fields whose root discriminant is larger than five times the present bound under assumption of the generalized riemann hypothesis the structure of this paper is as follows section 1 contains our results and section 2 lists the theorems used in the proofs the proofs themselves are largely suppressed received october 24 1980 revised december 24 1981 1980 mathematics subject classification primary 12 04 12a35 12a55

irregular primes to one million using fast algorithms for power series inversion based on the fast fourier transform and multisectioning of power series we have calculated all irregular primes up to one million including their indices of irregularity and associated irregular pairs using this data we verified that fermat s last theorem and vandiver s conjecture are true for these primes two primes with index of irregularity five were already known we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six a pair of integers p k is said to be an irregular pair if p is a prime k is an even integer satisfying 2 p are known see 8 our calculations show that vandiver s conjecture is true for all p 106 the table of irregular pairs could also be used to calculate iwasawa invariants for the corresponding primes we did not do this as part of our calculations see 9 or 4 for a discussion of this problem previous computations of irregular pairs have used algorithms that take 0 p2 arithmetic operations for each prime p the bernoulli numbers are defined in received by the editor april 30 1991 and in revised form september 10 1991 1991 mathematics subject classification primary 11b68 secondary 1 1d41 65t20 1 1y99 the first author was partially supported by national science foundation grant dms 9012989 d 1992 american mathematical society 0025 5718 92 1 00 25 per page

the calculation of a large cubic class number with an application to real cyclotomic fields the class number of the cubic subfield of the p th cyclotomic field is calculated for the prime p 11290018777 this is used to construct an example where the class number of the p th real cyclotomic field is larger than p in a recent paper 1 it is shown that if the generalized riemann hypothesis holds then hp7 p for the prime p 11290018777 here hp is the class number of q cos 27 p the maximal real subfield of the pth cyclotomic field in the present note we show that the grh is not needed so the result holds unconditionally it is a conjecture of vandiver with important consequences for fermat s last theorem and the theory of cyclotomic fields that p never divides hp it had been suggested that perhaps hp 1 014p as desired if the tail of the euler product had been ignored we would have obtained h3 6 1954 x 106 h2h3 1 473p we now give a different method for evaluating h3 which takes longer but has the advantage of giving the exact value and avoiding the grh as in 6 we may write vl l x wjk n f et t dt x n 2 j etdt n 1 nan1 na nviawhere a ir p and wx p2 2 x b e2xi b p sot wj 1 if we sum to n n in each sum then the error on the right is at most e a a2na3 cf 6 received november 29 1982 1980 mathematics subject classification primary 12a35 12a50 research partially supported by nsf research supported by nserc grant number a7649 1983 american mathematical society 0025 5718 82 0000 1367 01 75 303 this content downloaded from 157 55 39 207 on thu 20 oct 2016 04 14 55 utc all use subject to http about jstor org terms 304 eric seah lawrence c washington and hugh c williams there are two choices for x for definiteness choose x such that x b 1 e2w 3 e4 ril3 if bp 1 3 1 7526643766 3763375010modp respectively this is the cubic residue character modulo the prime ir 106256 3e2x 3 standard formulas on gauss sums 2 p 115 yield wq3 106256 3e2 3 p 1 2 this yields three choices for wx hence three possibilities for h3 however benedict gross suggested that perhaps if the error terms were made small enough in the above sums only one of these would allow h3 to be an integer this turns out to be the case writing the above formula for p l l x as wx 2 we have using n 250000 terms 56 437220121237002544 119 53491341223978301i 2 57418 064811064652753 4731 1007581056751240i with an error in i wx l 12 1 of at most 2 4 x 10 5 this yields the following possible values for i jpl i x l 57678 8276378680269 57678 8886140727151 57480 4620277658337 therefore h3 p i l 1 x 12 4r has the following possibilities respectively 6209212 00 6209225 13 6166576 73 the error is less than 10 2 so we must have h3 6209212 4 x 223 x 6961 this yields that h p is a multiple of p h2h3 2685 x 6209212 16 671 734 220

on the computation of class numbers of real abelian fields in this paper we give a procedure to search for prime divisors of class numbers of real abelian fields and present a table of odd primes 10000 not dividing the degree that divide the class numbers of fields of conductor 2000 cohen lenstra heuristics allow us to conjecture that no larger prime divisors should exist previous computations have been largely limited to prime power conductors

power integral bases in a parametric family of totally real cyclic quintics we consider the totally real cyclic quintic fields k n q v n generated by a root v n of the polynomial f n x x 5 n 2 x 4 2n 3 6n 2 10n 10 x 3 n 4 5n 3 11n 2 15n 5 x 2 n 3 4n 2 10n 10 x 1 assuming that m n 4 5n 3 15n 2 25n 25 is square free we compute explicitly an integral basis and a set of fundamental units of k n and prove that k n has a power integral basis only for n 1 2 for n 1 2 both values presenting the same field all generators of power integral bases are computed