Vol. 15, No. 3, 1965

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A remark on the lemma of Gauss

Fred Krakowski

Vol. 15 (1965), No. 3, 917–920

Let R be the ring of integers of some algebraic number field K and P = R[x0,,xr,y0,,ys], where the xi’s and yj’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=0rxiti and g(t) = j=0syjtj. According to the so-called Lemma of Gauss, I is equivalent to the product J of the ideals (x0,,xr) and (y0,,ys).

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
Received: 2 June 1964
Published: 1 September 1965
Fred Krakowski