• The Groups of a Field

    The following post is a digested version of a question I asked on math.SE a few months ago. ¶ To every field , we can associate two natural groups. These are the additive group and the multiplicative group of units . A fun question to ask, especially of someone just getting started on basic group and ring theory, is whether or not these two groups are ever isomorphic for any field. ¶ If you haven’t seen this question before, feel free to try it yourself!..

  • Prime Filters in Distributive Lattices II

    Recall from PFDL I, I introduced distributive lattices and filters, and we proved the easy direction of a characterization of Boolean algebras. Today I’ll detail a proof of the tougher and far more obscure converse—it involves some sneaky technology from formal logic. ¶ Theorem 1 states that, in a Boolean algebra, every (nonempty) prime filter is an ultrafilter. Its converse is as follows:..

  • Prime Filters in Distributive Lattices I

    I’d like to talk about some results pertaining to distributive lattices. In particular, there’s this one interesting theorem about Boolean algebras I’ve been thinking about lately. One direction is reasonably famous, pretty useful and not very hard to prove, so I’ll cover that. But what I really wanna talk about is the converse direction, which is a result that almost nobody I know has ever heard of, and is impossible to find anything about on the internet...

  • Why is a group?

    Often when people talk about groups, they say something like: groups are objects that encode the notion of symmetry. After working a bit with groups and group actions, it’s easy to convince yourself this is the case, but this sort of a posteriori explanation might seem a little circular—at least, it does to me...

  • Review of Integrals

    Mathematics has a lot of integrals in it. ¶ Some of them are kinda hard to compute, like , where you have to use integration by parts and rearrange or something. Some of them are really hard to compute, like , where you have to perform many unintuitive substitutions in sequence to obtain an unenlightening and kind of disgusting answer...

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