Hilbert Basis Theorem

David Hilbert is considered to be one of the greatest mathematicians of modern times. Attached to his name are numerous theorems, and his list of 23 problems presented in International Congress of Mathematics in 1900 has been very influential. I recently learned the following elegant proof of Hilbert Basis Theorem in an algebra course. Here are my notes taken in class that leads to the statement and proof of Hilbert Basis Theorem. According to Rotman’s Advanced Modern Algebra this proof is due to Sarges. The original proof of Hilbert is different, and apparently more complicated. In the same book, Rotman gives the following anectode which I found interesting. Here it goes:

The following anectode is well known. Around 1890, Hilbert proved the famous Hilbert Basis Theorem, showing that every ideal in \mathbb C[x_1,x_2,...,x_n] is finitely generated. As we will see, the proof is nonconstructive in the sense that it does not give an explicit set of generators of an ideal. It is reported that when P. Gordan, one of the leading algebraists of the time, first saw Hilbert’s proof, he said, “This is not Mathematics, but theology!” On the other hand, Gordan said, in 1899 when he published a simplified proof of Hilbert’s Theorem, “I have convinced myself that theology also has its advantages.”

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