Dirichlet asked whether there exist infinitely many real quadratic fields with 1 as relative class number. The order of the ideal class group is defined as the class number of the number field. Parker proved the existence of a particular class of real quadratic fields with relative class number 1 (Furness and Parker [2012]).
The first part of the chapter briefly discusses the preliminary conditions regarding class number and class group in a number field. Parker (Furness and Parker [2012]) investigated the relative class number of a real quadratic fieldQ(√ . m) by considering the continued fractional representation of√. The relative class number of K for a conductor f is the ratio Hd(f) between the class numbers Of = Z+fOK and OK.
The existence of a conductor of relative class number 1 can be proven by the following approach. Thus, we can establish the existence of a conductor of relative class number 1 for a real quadratic field Q(√ .m) when. In this section we give a construction for an infinite family of real quadratic fields of relative class number 1.
We conclude the chapter by relating the Mersenne primes to the relative class number of a real square field. Although Dirichlet could not find an answer to it, he obtained a very useful formula for the relative class number of a number field in terms of the fundamental unit.
Fundamental unit of norm − 1
Therefore, we have the desired result. m mod pOK, Asξm is the unit in OK, the lemma follows. We can assume that the basic units are ξm ∈Z[√. We now obtain the following corollaries. i) If p ≡ 1 mod 4 is an odd prime number that does not divide m, then the relative class number for the conductor p is not 1. ii) If p ≡3 mod 4 is an odd prime number that does not divide m, then the relative class number for the conductor p is odd. To prove the second part of the above corollary, we note that whenever the relative class number of the real square field Q(√ . m) is even for some normal p, θ(p) will always divide p−.
Proof: If the basis unit Q(√ . m) has norm −1, then −1 will be a quadratic remainder modulo any odd prime that divides d. Proof: If m is a prime congruent to 1 to 4, then it is an easy exercise to show that the fundamental unit Q(√ . m) has norm−1. If d is a square non-residue modulo a Mersenne prime f, then the relative class number of conductor f is 1.
A prime number is said to be a “Sophie Germain prime of the first kind” if 2f+1 is also prime. If d is a quadratic residue modulo 2f + 1, where f is a sufficiently large Sophie Germain prime of the first kind, then the relative class number for the conductor 2f + 1 is 1. Suppose f is a Sophie Germain prime, such that d is a quadratic residue modulo the prime numbers 2f+ 1 and 2f+ 1 do not share ˜α0β˜0.
Suppose that Q(√ . m) has only finitely many prime numbers of relative class 1. i) has only finitely many prime Mersenne numbers with d as a quadratic non-residue. ii) there are only more or less Sophie Germain primes of the first kind with d as a quadratic residue.
A Criterion
For any non-prime conductor f, our theorem follows from the fact that Hd(g) divides Hd(f) as g divides f (see Cohn [1962]). Dirichlet's unit theorem asserts that the group of units in the ring of integers of a pure cubic or real quadratic field is of rank one, and the smallest unit > 1 is referred to as the fundamental unit. In this chapter we consider a pure cubic field K = Q(√3 . m) with a power integral basis where m denotes a natural number.
In this chapter we investigate the divisibility and congruence properties of x, y and z with respect to the prime number 3 and. As a result, we derive the following results, the first of which agrees more simply with an old result of Gerth (Gerth et al. [1976]). When m = 3p is a square-free integer for some prime number p and does not divide 3hm, we show that the fundamental unit satisfies the congruences given in the theorem below.
We apply the same approach to study the congruence relations satisfied by the fundamental unit of a quadratic real field of class odd. An immediate consequence of our approach is the well-known classical result that a real quadratic field with discriminant having more than or equal to three prime factors has even class number (see Corollary 5.3.2). Using our approach, we can prove in an elementary way the following congruences which are given in (Zhang and Yue [2014]).
Divisibility of Class Number by 3
The next lemma follows from the assumption that 3 does not divide the class number of Q(√3 . m). Therefore, the unit u can be taken as ξmj for either j = 1 or j = 2 by modifying the element α appropriately, and the lemma follows. Now we are able to prove the following lemma, which is a crucial step in our proof of Theorem 5.1.1.
If the class number of K is not divisible by 3, then m is either a prime number or a multiple of 3. Since the class number of K is assumed to be coprime to 3, lemma 5.2.2 applies for such m with branched primes 3 and p. Then we can from the expression of x and use the fact that x∈Z say 3 divides each of a, b, c since 3 does not divide m.
We now consider the necessary congruence relations satisfied by the basic unit K = Q(√3 . p) when the class number is not divisible by 3. For other prime numbers p such that Q(√3 . p) has a power integral basis , we have p≡ 2.5 mod 9 and we get the following analogous statement as in Proposition 5.2.6.
Real quadratic fields with odd class number
The following result is classically known, but it can also be derived from the second relation in Lemma 5.3.1. If K = Q(√ . d) is a real square field with discriminant that has at least three prime factors, then the class number K is even. If ξd denotes the basic unit K and the class number K is not divisible by 2, then we can write ξd= αp2 = βpq2 according to Lemma 5.3.1.
We conclude by showing that the congruence relations stated in Theorem 5.1.3 follow directly from Lemma 5.3.1. But if the second, bare also thenx andy are likewise, contradicting equality in the norm equation x2−dy2 = ±1.
Examples
We verify that the fundamental unit ξm =x+yt+zt2 indeed satisfies the congruences in Theorem 5.1.2. We verify that when hm is not divisible by 3, the fundamental unit ξm =x+yt+zt2 of norm 1 satisfies the congruences in Theorem 5.2.8, or more precisely, in Theorem 5.2.7. As we mentioned earlier, the class group of a number field K measures how far the ring of integers is from a unique factorization into irreducible elements.
While the class group is defined as the quotient of the group of all fractional ideals of K by the subgroups of principal fractional ideals, it can also be seen as the Galois group of the maximum unbranched abelian extension of K by class field theory. Sato constructed quadratic number fields with a class number divisible by 5 based on elliptic curves in (Sato et al. [2011]). Lemmermeyer (Lemmermeyer [2013]) showed a method for constructing an unbranched quadratic expansion of cubic fields using points on suitable elliptic curves.
Taking our inspiration from (Lemmermeyer [2013]), we explicitly construct an undivided quadratic extension for every biquadratic field in an infinite family which originates from a non-twisting rational point on an elliptic curve chosen so suitable. Our results give unbounded quadratic extensions of infinitely many biquadratic fields Q(√ .. r,√ . m) where, if m is chosen appropriately, both r and m will be composite or neither r nor m will be a prime congruent to 1 mod 4. 3) which are composite numbers without squares. As a consequence, we actually obtain an alternative and constructive proof for the existence of infinitely many even-numbered biquadratic fields of class Our main result can be stated as follows.
Extension from a Non-torsion Point
So the only possibility for a common prime divisor of the numerator and denominator of x(2P) is p= 3 and in that case 3 divides s. But r and s are coprime so p= 3 can be the only common prime divisor for the numerator and denominator of y(2P), and in that case 3 divides s. Therefore, when 3 - s the fractions on the right-hand side of (6.3) are in their reduced form.
If we allow to be a square, then our construction gives an unramified quadratic expansion of the quadratic field Q(√ . m) under an additional condition (0< s < m). We can derive a corollary from the above lemma in the case where the integer m is a positive multiple of 3. Thus, in this section we have explicitly constructed an everywhere unchanged quadratic expansion of a biquadratic field that we associate with a non-torsion relation - endpoint on the elliptic curveEm where the coordinates of the point satisfy certain light conditions.
In this section we show that we can start with a point without rotation P0 and repeat the procedure of the previous section for any multiple Pi = 2i(P0) = (r. We have studied the relation between the point on an elliptic curve and the class number of field of numbers generated by that point. Keeping her work in mind, we are trying to relate the divisibility of the class number of a real quadratic field by 3 to the existence of a point of infinite order on a suitable elliptic curve.
For each field of the completely real number k and for each prime number p, the Iwasawa invariants λp(k) and μp(k) capture the increase of the p-part of the class number as we go along the tower of fields in the zygotomic Zp. -expansion of k. The disappearance of these invariants means that the p part of the class number eventually stabilizes as we go along the tower of fields in the cyclotomicZp-zp extension. Furthermore, we are trying to derive some easily verifiable consequences of the Greenberg conjecture in terms of the coordinates of points on a suitable elliptic curve.
Goldfeld, D.: 1985, Gauss class number problem for imaginary quadratic fields, Bulletin of the American Mathematical Society. M.: 1976, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze.
An Infinite Family
