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## Main Question or Discussion Point

Does the notion of continuity exist in modern algebra?

If so how do they arise?

If so how do they arise?

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Does the notion of continuity exist in modern algebra?

If so how do they arise?

If so how do they arise?

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Hurkyl

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mathwonk

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see zariski topology, and the metric completion of a ring wrt an ideal.

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So it can happen in algebra but rare for pure algebra as opposed to algebra mixed with other maths disciplines like topology.

But would you say 'discreteness' plays a large role in modern algebra?

How would it sound if someone said they like modern algebra because of its discreteness which lacks in say analysis and topology?

But would you say 'discreteness' plays a large role in modern algebra?

How would it sound if someone said they like modern algebra because of its discreteness which lacks in say analysis and topology?

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Hurkyl

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mathwonk

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then in this metric, where m = the maximal ideal of the origin in the polynomial ring, a formal power series is the limit of the sequence of its partial sums. i.e. one can complete a polynomial ring wrt a maximal ideal, and get a power series ring.

this leads to the concept of the completion of a ring wrt an ideal. this concept helps recover local concepts in algebra,

"more local" than given by the zariski topology.

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matt grime

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mathwonk

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but one can also look at more complicated invariants such as the associated affine scheme, or in the case of a local ring, its completion wrt the metric defined above.

for the local ring of a smooth point on an algebraic variety this leads to a proof of the theorem in pure algebra that the ring is a ufd because its completion is so.

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HallsofIvy

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So is that why Herman Weyl said something like "...algebra and topology fight for every branch of mathematics." So even a fundalmental discipline like analysis is a combo of algebra and topology.

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I can't define pure algebra just as I can't define mathematics. But I can give some examples. Galois theory is pure algebra. Whereas topics in algebraic topology is not pure algebra. There may also me things in between.

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What do you mean by invariants? Never changing?

but one can also look at more complicated invariants such as the associated affine scheme, or in the case of a local ring, its completion wrt the metric defined above.

for the local ring of a smooth point on an algebraic variety this leads to a proof of the theorem in pure algebra that the ring is a ufd because its completion is so.

http://en.wiktionary.org/wiki/invariant

Do you mean its properties?

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mathwonk

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matt grime

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The ultimate aim of mathematics for some people is to find a set of invariants that uniquely characterise objects up to whatever level of equivalence you are working with. See, e.g. compact surfaces, genus, homotopy, and the poincare conjecture, for example.

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Hurkyl

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The Galois group is profinite, its topology plays a criticial role in the theory.Galois theory is pure algebra.

Finite profinite groups are discrete -- if you've only worked with finite Galois extensions, that would explain why you haven't seen this before. A useful example of an infinite Galois extension is [itex]\mathbf{\bar{Q}} / \mathbf{Q}[/itex]: the algebraic numbers over the rationals.

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matt grime

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Whereas topics in algebraic topology is not pure algebra.

Why not? Yes, I am playing devil's advocate, but I'd like you to examine your own preconceptions about mathematics.

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mathwonk

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HallsofIvy

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Does the fact that he said "

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