Integrated Development of Algebra in Type Theory

We present the project of developing computational algebra inside type theory in an integrated way. As a rst step towards this, we present direct constructive proofs of Dickson's lemma and Hilbert's basis theorem, and use this to prove the constructive existence of Grr obner bases. This can be seen as an integrated development of the Buchberger algorithm, and so far we have a concise formalisation of Dickson's lemma in Half, a type{ checker for a variant of Martin-LL of's type theory. We then present work in progress on understanding commutative algebra constructively in type theory using formal topology. Currently we are interested in interpreting existence proofs of prime and maximal ideals, and valuation rings. We give two case-studies: a proof that certain a are nilpotent which uses prime ideals, and a proof of Dedekind's Prague theorem which uses valuation rings.