I suspect that this theorem is indeed due to Weil. 

"Foundations of Algebraic Geometry" by Weil was published in 1946, but the 1944 paper "[Some Properties of Ideals in Rings of Power Series][1]" by Claude Chevalley (Transactions of the American Mathematical Society, Vol. 55, No. 1 (Jan., 1944), pp. 68-84) attributes to Weil the development of the theory around "ideals in polynomial rings" over a decade earlier in "Arithmetique et geometrie sur les varietes algebriques" in 1935 (see footnote on p. 83).

Reading the [AMS review][2] it seems the only other possible originators would have been Siegel, or perhaps Noether or van der Waeden. I don't have a copy of Weil's 1935 work, but you might track it down and (if you can read enough French) check for this particular result.


  [1]: http://www.jstor.org/stable/1990140
  [2]: http://www.ams.org/journals/bull/1936-42-09/S0002-9904-1936-06368-8/S0002-9904-1936-06368-8.pdf