Math Aunty

I’ll skip the spiel about the blogs that I haven’t been posting and the empty promise of more frequent blogs and hop straight to it. In Abstract Algebra, we learn about groups and subgroups, rings and subrings, vector spaces and subspaces… however, we don’t learn about fields and subfields. I’ve always thought that was strange. I never thought to question it too hard because I was in my senior year of undergrad and my thought was ‘GTFO’… but today… today the mystery has been solved. This is for people who always wondered but didn’t care enough to ask why.

With groups and rings, we were interested in the different subgroups and subrings contained within fixed groups and rings. We classify these subgroups and subrings in many different ways (normal, proper, ideal, nontrivial, etc). However, once we start to work with fields, we talk about field extensions instead of subfields. We don’t talk about the rational numbers as a subfield of the complex numbers. We talk about the complex numbers as an extension of the rational numbers. Why? Because we fix a field and we’re studying the different ways we can add elements to that field to create new fields. Basically, with groups and rings we have H as a subgroup of G and we study the different H. With fields, we have a field F contained in a larger field E and we look at the different E. We build more complicated fields containing the field we know the most about (the rationals) until we get the complex numbers instead of starting with the complex numbers and looking at the fields contained in them. So yeah… This is short and sweet and a little clarifying but in a mostly useless way. 🙂