S. Chowla

The nonexistence of nontrivial linear relations between the roots of a certain irreducible equation

Abstract Let x1,…, xn be the roots of an irreducible equation of degree n over Q. Under what conditions can we have “nontrivial” relations a 1 x 1 + … + a n x 1 = 0 (a n ∈ Q)? In some cases it is shown that the nonexistence of nontrivial relations depends on the nonvanishing of Dirichlets series L(s, χ) at s = 1, when χ is a character with χ(−1) = −1.

Determination of the Stufe of certain cyclotomic fields

Abstract The authors prove that the Stufe of the cyclotomic field K = Q(e 2πi p ) , where p is a prime ≡ 7 (mod 8), is exactly 4.

On the number of zeros of diagonal cubic forms

Abstract Let M s , be the number of solutions of the equation X 1 3 + X 2 3 + … + X s 3 =0 in the finite field GF ( p ). For a prime p ≡ 1(mod 3), ∑ s=1 ∞ M s X s = x 1 − px + x 2 (p − 1)(2 + dx) 1 − 3px 2 − pdx 3 , M 3 = p 2 + d(p − 1) , and M 4 = p 2 + 6(p 2 − p) . Here d is uniquely determined by 4 p = d 2 + 27b 2 and d ≡ 1( mod 3) .

The least prime quadratic residue and the class number

For an odd prime p, let l(p) denote the least positive prime which is a quadratic residue modulo p. An elementary proof is given of the following: For p ≡ 3 mod 8 and p > 3, l(p) = (p + 1)4 iff the class number h(−p) = 1.

On a theorem of Müller and Carlitz

Let r(n) denote the number of representations of n as a sum of two squares. We have ∑n⩽χ r(n) log(x/n) = Ax +B log x + C + O(x−14) where A, B, C are constants. The evaluation of C in “closed form” is established by a new method in this paper.

On the integer points on some special hyper-elliptic curves over a finite field

Abstract If l r(p) is the least positive integral value of x for which y2 ≡ x(x + 1) ⋯ (x + r − 1)(modp) has a solution, we conjecture that l r(p) ≤ r2 − r + 1 with equality for infinitely many primes p. A proof is sketched for r = 5. A further generalization to y2 ≡ (x + a1) ⋯ (x + ar) is suggested, where the as are fixed positive integers.

Congruence properties of class numbers of quadratic fields

Abstract The authors prove that the class number of the quadratic field Q(√−g) is divisible by 3 if g is a prime of the form 27n2 + 4.

A note on the Hecke hypothesis and the determination of imaginary quadratic fields with class-number 1

Abstract The authors propose a hypothesis whose proof would provide, in particular, a solution of the “class-number 1” problem, recently solved by Stark and Baker.

A note on the hypothesis that L(s, χ) > 0 for all real non-principal characters χ and for all s > 0

Abstract If χ(n) is an odd real character (mod k), it is an unsolved problem whether ∑ n=1 ∞ x(n) n s > 0 (s > 0) . We propose a hypothesis which ensures the truth of this inequality.

Remarks on Equations Related to Fermat’s Last Theorem

For odd k, define θ(k) as the least value of s such that has a non-trivial Solution over the integers. Fermat’s Last Theorem impl ies that θ(k) > 3 for odd k > 3.