© Hindawi Publishing Corp. CONTINUED FRACTIONS AND CLASS NUMBER TWO

We use the theory of continued fractions in conjunction with ideal theory (often called the infrastructure) in real quadratic fields to give new class number 2 criteria and link this to a canonical norm-induced quadratic polynomial. By doing so, this provides a real quadratic field analogue of the well-known result by Hendy (1974) for complex quadratic fields. We illustrate with several examples.


Introduction.
In [4], we gave criteria for real quadratic fields of ERD-type (those with discriminant of the form ∆ = 2 + r with r | 4 ) to have class number 2 in terms of a canonical quadratic polynomial, thus providing a real quadratic field analogue of a result of Hendy [2] for complex quadratic fields.Herein we give more general criteria for (not necessarily ERD type) real quadratic fields in terms of the same polynomial using the theory of the infrastructure.Moreover, we are able to handle certain ERD types that were not covered in [4].

Notation and preliminaries.
In this section, we see how ideal theory is linked to continued fractions via what the late Dan Shanks (see [6]) called the infrastructure of a real quadratic field.
First we need the following notion.
Definition 2.1.Suppose that D ∈ N is not a perfect square.Then a quadratic irrational is a number of the form where Q = 0 and P 2 ≡ D (mod Q).Also, the algebraic conjugate of α is We are interested in linking this notion to both continued fractions and ideals.To establish the link with continued fractions, we first note that it is well known that a real number has a periodic continued fraction expansion if and only if it is a quadratic irrational (see [7,Theorem 5.3.1,page 240]).Furthermore a quadratic irrational may have a purely periodic continued fraction expansion which we denote by α = q 0 ; q 1 ,q 2 ,...,q −1 , (2.3) meaning that q n = q n+ for all n ≥ 0, where = (α) is the period length of the simple continued fraction expansion.It is known that a quadratic irrational α has such a purely periodic expansion if and only if α > 1 and −1 < α < 0. Any quadratic irrational which satisfies these two conditions is called reduced (see [7,Theorem 5.3.2,page 241]).Now we are in a position to bring in the theory of ideals and link the three together.
Let D > 1 be a squarefree positive integer and set Define The value ∆ is called a (fundamental) discriminant or (field) discriminant with associated radicand D, and ω ∆ is called the (fundamental) principal surd associated with ∆.
For simplicity, we will refer to ∆ as a discriminant with associated radicand D and principal surd ω ∆ .This will provide the canonical basis element for certain rings that we now define.
It may be shown that any Z-module where a, c ∈ N with 0 ≤ b < a.We will only be concerned with primitive Z-modules, namely those for which c = 1.In other words, I is a primitive Z-submodule of O ∆ if whenever I = (z)J for some z ∈ Z and some Z-submodule Now we set the stage for linking ideal theory with continued fractions by giving a criterion for a primitive Z-module to be a primitive ideal in O ∆ .A nonzero Z-module I as given in (2.7) is called a primitive O ∆ -ideal if and only if P 2 ≡ D (mod Q) (see [7, Theorem 3.5.1,page 173]).Henceforth, when we refer to an O ∆ -ideal it will be understood that we mean a primitive O ∆ -ideal.Also, the value Q/σ is called the norm of I, denoted by N(I).Hence, we see that I is an O ∆ -ideal if and only if α = (P + √ D)/Q is a quadratic irrational.Given the notion of a reduced quadratic irrational discussed earlier, it is not surprising that we define a reduced ideal I to be one which contains an element β = (P + √ D)/σ such that I = [N(I), β], where β > N(I) and −N(I) < β < 0, since this corresponds exactly to the reduced quadratic irrational α = β/N(I) > 1 with −1 < α < 0. In fact, the following theorem holds.Theorem 2.2.Let ∆ be a discriminant with associated radicand D.
Also, it is not surprising that we define the conjugate ideal of I to be Now we link continued fractions to the ideals defined above.Let I be an O ∆ -ideal given by (2.7).Define P 0 = P , Q 0 = Q, and recursively for j ≥ 0, (2.9) (2.10) It follows that we have the simple continued fraction expansion of α given by Now the stage is set for the appearance of the result that formally merges ideals and continued fractions.We only need the notion of the equivalence of two O ∆ -ideals I and J, denoted I ∼ J to proceed.We write I ∼ J to denote the fact that there exist nonzero integers α, β ∈ O ∆ such that (α)I = (β)J, where (x) denotes the principal O ∆ -ideal generated by x ∈ O ∆ .For a given discriminant ∆, the class group of O ∆ determined by these equivalence classes, denoted by C ∆ , is of finite order, denoted by h ∆ , called the class number of O ∆ .One result on the class group that we will need in the next section is the following.Theorem 2.4.Suppose that ∆ ∈ N is a discriminant, P j , Q j are given by (2.10) and (2.12) for nonnegative j ∈ Z. Then I 1 ∼ I j for all j ∈ N. Furthermore, there exists a least natural number n such that I n+j is reduced for all j ≥ 0, and these I n+j are all of the reduced ideals equivalent to I 1 .If ∈ N is the least value such that I n = I +n , then for j ≥ n − 1, α j = (P j + √ D)/Q j all have the same period length = (α j ) = (α n−1 ).
From the continued fraction algorithm, we see that if represents the norms of all reduced ideals equivalent to I.This is achieved via the simple continued fraction expansion of α = (P + √ D)/Q.Two important results needed in the next section are given as follows.
Theorem 2.5 also has relevance to the solution of quadratic Diophantine equations (see [9]). 3. Class number two results.In this section, we use the following notation for our canonical polynomial: , then p is some Q j /2 for some j with 1 ≤ j ≤ (α) = , where either α = ω ∆ or α = (b q +ω ∆ )/q, where the ideal ᏽ = [q, b q + ω ∆ ] is the O ∆ -prime above q.
Recently in [1], Andrew Granville, the author, and Hugh Williams were able to unconditionally verify a conjecture made at the end of Chapter six of [5], which had theretofore only been known to hold under the conditions mentioned in Remark 3.10.It is, however, unlikely that those techniques can be used to unconditionally verify Conjecture 3.9.