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.

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**Edit:** For remarks which are perhaps related/interesting (in terms of Weil's background and his familiarity with Kronecker's work) read from the last paragraph of page 12 [here][3] and the referenced [ICM address][4] by Weil in 1950.


  [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
  [3]: http://books.google.com/books?id=YLcPxfZW47EC&lpg=PA13&ots=M9cEaHHZzY&dq=%2522field%2520of%2520definition%2522%2520kronecker&pg=PA12#v=onepage&q&f=false
  [4]: http://www.mathunion.org/ICM/ICM1950.2/Main/icm1950.2.0090.0102.ocr.pdf