Solving several important problems in the theory of invariants. Hilbert's basis theorem solved the principal problem in the 1800s invariant theory by showing that any form of a given number of variables and of a given degree has a finite, yet complete system of independent rational integral invariants and covariants.
Unifying the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers").
His suggestion in 1920 that mathematics be formulated on a solid logical foundation (by showing that all of mathematics follows from a system of axioms, and that that axiom system is consistent). Unfortunately, Gödel'sIncompleteness Theorem showed that his grand plan was impossible.
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