RESEARCH EXPOSITORY AND SURVEY ARTICLES PRIME VALUED POLYNOMIALS AND CLASS NUMBERS OF QUADRATIC FIELDS

. It is the purpose of this paper to give a survey of the relationship between the class number one problem for real quadratic fields and prime-producing quadratic polynomials; culminating in an overview of the recent solution to the class number one problem for real quadratic fields of Richaud-Degert type. We conclude with new conjectures, questions and directions.


INTRODUCTION
There has long been a certain fascination with prime valued polynomials.For example in 1772 Euler [1] found that x =x + 41 is prime for all integers x with 1 <_ x <_ 40.Later we will see that this is not so much a property of the polynomial as it is of its discriminant -163.
It happens that the more general polynomial q(x) a Dedekind domain every non-zero fractional ideal may be uniquely written as a product of powers of distinct prime ideals.Hence the monoid, of non---zero fractional ideals of a Dedekind domain is a group.The principal fractional ideals (i.e., those ideals with a single generator) form a subgroup P of I.The quotient group I/P is called the ideal class group of the Dedekind domain.It is a fact that rings of integers O K of number fields K are Dedekind domains.We let C K denote the ideal class group of O K (or simply of K).Dirichlet proved that C K is finite.We refer to its order h K as the class number of K. Moreover we see that O K is a principal ideal domain (P.I.D.) if and only if h K 1.It is a well-known fact that O K is a unique factorization domain (U.F.D.) if and only if it is a P.I.D. Thus Kummer's essential obstruction in his investigation of Fermat's last theorem were cyclotomic fields with class number bigger than one.In fact Fermat's last theorem is true for a prime p > 2 if p does not divide the class number of the p-th cyclotomic field Q((p), (where (:p is a primitive p-th root of unity).Class numbers bigger than one somehow measure how far away O K is from being a U.F.D.
QUESTION: If h k > h L then does this mean that K is farther away (in some sense) from being a UFD then L is?In his survey article [2] Masley cites well-known examples of, Furtwangler to conclude in the negative and says: "The meaning of class numbers larger than 2 is then a complete mystery".However the Furtwangler examples look at "activity' in the Hibert Class field.In fact, in response to a problem stated by Narkiewicz in 1974 (to arithmetically characterize all algebraic number fields with class number bigger than 2), David rush solved the problem in terms of elementary factorization properties in 1983.The result is too technical to state here but a result of U. Krause for the cyclic case shows the flavour of the approach as follows.(The term x primary should be understood to mean xlyz implies xly or xlz n for some integer n _> 1).
THEOREM.C K is cyclic of prime power order if and only if there exists on m _> 0 such that the m th power of every irreducible integer is a product of at most m primary integers, h K is given by the smallest possible m.
Another result (attributable to Narkiewicz) which examines the h K > h L phenomenon is as follows.
Let FK(X be the number of non-associated integers a of K with unique factorization and INKIQ(a) _( x.Then we have h K > h L if and only if FK(X)/FL(X 0. In F. If AK(X log(FK(x)/log x) then lira (AK(x)/log log x) 1 (1/hg). x-(R) CONCLUSION.The answer to the Question depends upon what you mean by "farther away from".The answer is clearly "yes" in terms of elementary factorization criteria such as that of Kranss (above) or D. Rush; or in terms of "density" as with FK(X and AK(X as above.The answer is not so clear if you look outside K in terms of the principal ideal theorem of class field theory as Masley interpreted the Furtwangler examples.

2.
COMPLEX QUADRATIC FIELDS AND PRIME QUADRATICS Heilbronn and Linfoot [3] proved that there are at most ten complex quadratic fields with class number one; namely Q(]-for d E {1,2,3,7,11,19,43,67,163, and possibly one other}.Baker [4] and Stark [5] independently eliminated the other potential d.For a complete survey of the solution to the class number one problem see Goldfeld [6].Also included therein is the history of the solution to a more general problem going back to Guass (i.e., to give an effective lower bound for discriminants of all complex quadratic fields having a given class number).The 1987 Cole prize in number theory was jointly awarded to D. Goldfeld, B. Gross and D. Zagier for their solution to this problem (see [7, pp.232-234]).Now we return to the prime quadratics introduced earlier.In 1913 Rabinovitch obtained: THEOREM 2.1.(Rabinovitch [8] and [9]).Let d 3 (mod 4), d > 0 and K q(--.Then p(x) xx + (d + 1)/4 is prime for all integers x with 1 _< x <_ (d-3)/4 if and only if h K 1.
We now see the reason for the comment at the outset of the article that this is not property of the quadratic but of its discriminant -163.For an interesting, (albeit older) note on the subject see Lehmer [10].We now turn to the relationship between class number 2 for complex quadratic fields and certain prime quadratics.Baker [11] and Stark [12] proved that there are exactly eighteen complex quadratic fields Q(-- They occur for d {5,6,10,13,15,22,35,37,51,58,91,115,123,187,235,267, 403,427}.Prime quadratics in relation to complex quadratic fields of class number 2 were discovered by Hendy [13] where p is an odd prime then h K 2 if and only if f(x) 2x + p is prime for all x with 0 _< x < (II prime for all integers x with 0 _< x < (/(p-1)/2.
We note that from the genus theory of Gauss h K 2 for K Q(/') if and only if d is one of types I-III in Theorem 1.2.Hence Theorem 1.2 together with the solution to the class number two problem for complex quadratic fields yields the next remarkable relationship with prime quadratics.
(P2) If p is an odd prime then f(x) 2x + p is prime for all integers x with 0 _< x < / if and only if p E {3,5,11,29}.
As an illustration of the above we have: EXAMPLE 2.2.2x + 29 is prime for all integers x e with 0 _< x <_ 14.EXAMPLE 2.3.2x + 2x + 19 is prime for all integers x with 0 <_ x _< 3. EXAMPLE 2.4.7x + 7x + 17 is prime for all integers x with 0 5 x <_ 5.
There is only one other class of fields for which there is a complete answer to the class number two problem.There are exactly two cyclotomic fields K Q(n (where n $ 2 (mod 4)) such that h K 2. They occur for n 39 and 56, (see Masley  [14] for a survey of small class groups for abelian number fields.) The solution of the class number one and two problems for complex quadratic fields led to a neat set of solutions (P1)-(P4), for related prime quadratics.The story is not so complete for real quadratic fields as we will see in the next section.

REAL QUADRATIC FIELDS AND PRIME QUADRATICS
The complete solution given in 2 does not yet have an analog for real quadratic fields.This is true because, at this juncture in mathematical history very little is known about class numbers of real quadratic fields.For example it is still not known whether there exist finitely many real quadratic fields with a given class number.In particular an open conjecture of Gauss says that there are infinitely many real quadratic fields with class number one.As Goldfeld said in his response to the receipt of the Cole prize (op.cit.)-"This problem appears quite intractable at the moment."In point of fact we do not yet know whether there are infinitely many number fields with class number one.However with respect to prime quadratics some progress has been made.We begin with the introduction of a restricted class of real quadratic fields, which have been a topic of interest from several perspectives for some time.

If d
/ + r where !> 0 is an integer and r divides 4l with integer r such that -! < r <_ l, then Q(rd) (or simply d) is said to be of Richaud-Degert type (or (R-D)-type), (see [15] and [16]).If Irl e {1,4} then d is said to be of narrow (R-D)-type.In the general case they are called (wide) (R-D)types.In [17], S.
Chowla conjectured that primes p of narrow (R-D)type / + 1 with l > 26 satisfied h(p) > 1, where h(p) h K for K Q(q).Several attempts have been made at solving this conjecture, and we now wish to link this investigation with our search for prime quadratics.
A step toward a real analog of Rabinovitch's Theorem is the following result of Kutsuna [18]: If d 1 + 4m is square-free and -x + x + m is prime for all integers x with 1 < x < .then h(d) 1.   Kutsuna's result however, is incomplete in that it does not give necessary and sufficient conditions for h(d) 1 in terms of prime quadratics.Moreover it fails for some of the most interesting fields.For example it fails when m is a square thereby eliminating Chowla's conjecture.In an attempt to link the Chowla conjecture to a search for a real analogue of Rabinovitch's result Mollin  [19] discovered the following pleasant connection: THEOREM 3.2.Let d 4m+ 1 be square-free where m is a positive integer.Then the following are equivalent.
(II) p is inert in Q(d) for all primes p < m; (i.e. (d/p) -1 for all odd primes p < m, where ] is the Legendre symbol; and m is odd).
(III) f(x)=-x + x + m 0 (mod p) for all positive integers x and primes p satisfying x < p < m. (IV) f(x) is prime for all integers x with 1 < x < m.
Note that it is known from more general results proved in Mollin [20] that if d / + 1 > 17 is square-free and t# 2q for an odd prime q then h(d) > 1, (see also [21]).The further reduction to 4q + 1 being a prime is known by the genus theory of Gauss.In fact the reduction to t 2q, q > 2 prime is known (eg.see [17, p.48]).In [19] however all such reductions are accomplished via elementary arithmetic techniques.
In any case the Chowla conjecture and the above yield the following conjectures.
The overriding assumption in the conjectures is that p --4q + 1 is prime and q > 2 is prime.CONJECTURE 3.1.-x -t-x -I-q is prime for all integers x with 1 < x < q if and only if q <_ 13.CONJECTURE 3.2.
-x + x + q2 $ 0 (mod r) for all positive integers x and primes r satisfying x < r < q if and only if q <_ 13.Although the Chowla conjecture remains open for the above case, Mollin and Williams   [22] were able to prove it under the assumption of the generalized Riemann hypothesis (GRH); i.e., the Riemann hypothesis for the zeta function of Q().
Further investigations by Mollin [23] revealed the following result.In what follows (W + Vrd)/2 denotes the fundamental unit of Q(]d) and N((T + U/d)/2) where N denotes the norm from Q(Fd) to Q.For convenience sake we let A (T--I)/U .
THEOREM 3.3.Let d 1 (rood 4) be a positive square-free integer such that (/d-I)/2 <_ A. Then the following are equivalent.(2) p is inert in Q(qrd for all primes p < A. (3) f(x) -x -b x -b (d-l)/4 0 (rood p) for all positive integers x and primes p satisfying x (4) f(x) is prime for all integers x with 1 < x < (fd-'/.. Theorem 3.2 is an immediate consequence of Theorem 3.3 as is the following result on the other narrow R-D types.COROLLARY 3.1.Let d m 4 > 5 be square-free.Then h(d) > 1 unless d 4p + 1 where p is prime, in which case the following are equivalent: (iii) f(x)--x / x / p 0 (rood q) for all positive integers x and primes q satisfying q < x < , , .
(iv) f(x) is prime for all integers x with 1 In [24] Yokoi conjectured thai; h(d) > 1 when d--q-b 4 is squarefree with q > 17 prime.Under the assumption of the generalized Riemann hypothesis this conjecture follows from the techniques used by Mollin and Williams in [22].Thus we have: CONJECTURE 3.4.If q is an odd prime then -x -b x / (q-b 3)/4 is prime for all integers with 1 < x ( q2 + 3)/2 if and only if q <_ 17.
In [25] Mollin and Williams were able to make substantial progress and found all real quadratic fields of narrow R-D type of class number one.To state the results we will label some conditions at this uncture since we will have oecassion to refer to them often.
What is most surprising and revealing is their next result.THEOREM 3.5.(Mollin and Williams [25]).If (III) holds for d >13 then d 1(rood 4) and d is of narrow R-D type.
In view of Theorem 3.5 we may now assert that Theorem 3.3 is es,entlally a sta(,ment about narrow R-D types.Theorem 3.5 was also the key for Molla and Wllams to find all real quadratic ficlds of narrow R-D type wth class number one.[25]).If the G.R.H. holds then (III) (IV) If and only if d is an entry on Table 2.1 COROLLARY 3.2.Assume the G.R. tl.holds.All real quadratic fields Q(.,/]) of narrow R-D type with h(d) 1 are fr d {2, 3,17,21,29,37,53,77,101,173,197,293, 437,677}. (Note that 5 is not generally considered to be an R-D type since it does not fit the fundamentM ut pattern.Morver 6,7,ll e de R-D types and 13 3 + 4 t + r is not R-D type siuce must be less than t).Thus (under GRH) Corollary 3.2 verifi conjectures of Chowla, Mollin and Yokoi (see [51).
Ts left on the problem for wide R-D types.Mollin and Williams were able to settle the question in [26].Moreover they discovered some strong connections between the class number one problem and prime producing quadratic polynomials.For example: THEOREM 3.7.
It was not ,lntil later ork in [27] wllcc Mollin and W:lliams were able to prove results similar to Theorem 3 7 for tl,,.renaining R-D t?pes.In [26] they made two c,,mjectu,..s concc,ning these R-I) types, which they were able to prove in even generality in [27].For example, t,h,,')-p,ov,:d the {ollowmg in [27].TIIBOREM 3.8.Let d pq, p < q where p 3 _--q (mod 4) are l;rimes and d _= ,5 (rood 8) If pxpxl(t,-q)/41 is t)r, me or 1 for all it, cgcts x with _< .< (fd/'l-1 / 2 then h(d)-: :1.0 Despite the seemingly more general nature of Theorem 3.8, the authors are couvinced that if the hypothesis of Theorem 3.8 holds !hen d s of R-D type.In [27] they showed that under the assumption of a suitable l(icrnann hypothesis the coxjectue holds.For similar related theorems and conjcctmes see [27]. In [26] Mollin and Williams wc[e able to ittvoke the generahzed Ricmann hypothesis (GRH) fcr the zeta-function of (:t) to find all eal quadratic fields of Ricbaud Degert type with h(d) 1.In [28] they were able to :,,n,)ve the C, RH and proved the following.
In what follows extended R--D fype I,,.'a,,s those forms d 12+ r with r dividing 4l.THEOREM 3.9.l/Vith possibly only on ,,ore value rernaint'g all real quadratic fields of Extended R-D type Q(/]) with h(d)