A remark on the lemma of Gauss

Fred Krakowski

Vol. 15 (1965), No. 3, 917–920

DOI: 10.2140/pjm.1965.15.917

Abstract

Let R be the ring of integers of some algebraic number field K and P = R[x₀,⋯,x_r,y₀,⋯,y_s], where the x_i’s and y_j’s are indeterminates. Call two ideals of P equivalent, if after substitution of the indeterminates by arbitrary elements of R they always yield identical ideals in R. For example, consider the ideal I generated by the coefficients of the product of the two polynomials f(t) = ∑ _i=0^rx_itⁱ and g(t) = ∑ _j=0^sy_jt^j. According to the so-called Lemma of Gauss, I is equivalent to the product J of the ideals (x₀,⋯,x_r) and (y₀,⋯,y_s).

The object of this note is to show that the ideal I has the following minimal property: It has the smallest number of generators, namely r + s + 1, among all ideals in P which are equivalent to J in the above sense.

Mathematical Subject Classification

Primary: 10.69

Milestones

Received: 2 June 1964

Published: 1 September 1965