Vol. 30, No. 2, 1969

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An explicit formula for the units of an algebraic number field of degree n 2

Leon Bernstein and Helmut Hasse

Vol. 30 (1969), No. 2, 293–365

An infinite set of algebraic number fields is constructed; they are generated by a real algebraic irrational w, which is the root of an equation f(w) = 0 with integer rational coefficients of degree n 2. In such fields polynomials Ps(w) = a0ws + a1ws1 + + as1w + as and

Qs(w) = b0ws + b1ws −1 + ⋅⋅⋅+ bs−1w + bs

( s = 1,,n 1;ak,bk rational integers) are selected so that the Jacobi-Perron algorithm of the n 1 numbers

Pn −1(w ),Pn− 2(w ),⋅⋅⋅ ,P1(w)

carried out in this decreasing order of the polynomials, and of the n 1 numbers

Q1(w ),Q2(w ),⋅⋅⋅ ,Qn− 1(w )

carried out in this increasing order of the polynomials both become periodic.

It is further shown that n 1 different Modified Algorithms of Jacobi-Perron, each carried out with n 1 polynomials Pn1(w),Pn2(w),,P1(w) yield periodicity. From each of these algorithms a unit of the field K(w) is obtained by means of a formula proved by the authors is a previous paper.

It is proved that the equation f(x) = 0 has n real roots when certain restrictions are put on its coefficients and that, under further restrictions, the polynomial f(x) is irreducible in the field of rational numbers. In the field K(w)n 1 different units are constructed in a most simple form as polynomials in w; it is proved in the Appendix that they are independent; the authors conjecture that these n 1 independent units are basic units in K(w).

Mathematical Subject Classification
Primary: 10.65
Received: 1 February 1968
Published: 1 August 1969
Leon Bernstein
Helmut Hasse