Another thing he did was unify the field of algebraic number theory. He also provided the first correct and complete axiomatization of Euclidean geometry. It replaced Euclid's axiomatization of geometry from his 1899 book. Hilbert has also laid the foundations of functional analysis by studying integral equations and formulating a first version, in terms of quadratic forms, in infinitely many variables...
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This came to be known as Hilbert space. This was also foundational for quantum mechanics in the 1920's. Hilbert was a very intelligent man and contributed too many things in math. One of the last things he did was put forth an influential list of 23 unsolved problems at the International Congress of Mathematics in Paris in 1900...
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