Talk:2028: Complex Numbers

I assume this is strictly a coincidence, but in reference to the title-text, I'll just mention that Caucher Birkar [the mathematician whose Fields Medal was stolen minutes after he received it in Rio de Janeiro on Weds (1Aug2018)] received the award for work in algebraic geometry. Arcanechili (talk) 16:34, 3 August 2018 (UTC)

• Perhaps it's causal not coincidental. Medal theives and perhaps Randall might read the news also. [[1]] 162.158.79.209 00:34, 4 August 2018 (UTC)

I've added a basic description of Abelian groups in the title text, and that's about as much as I know about such topics. I'm not sure what a "meta-Abelian group" is, is that an Abelian group of other groups? Also, could someone add basic descriptions of algebreic geometry and geometrical algebra? 172.68.94.40 18:42, 3 August 2018 (UTC)

In the title text, since groups are a concept within mathematics, it seems odd to consider mathematics as a whole forming any sort of group within itself, which I suspect is the first part of the pun. Secondly, since groups involve the commutative property, I think the last part is a pun about the order of the words algebra and geometry, as if they're commutative themselves! Ianrbibtitlht (talk) 19:19, 3 August 2018 (UTC)

I meant to say 'abelian' groups involve the commutative property, and the meta prefix is referring to the fact that it's about the names rather than the mathematical details - i.e. commutative in metadata only. Ianrbibtitlht (talk) 19:24, 3 August 2018 (UTC)

I guess the joke is that informally mathematicians form a group (a number of people classed together), what would strictly be a set in mathematics. While in mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies specific conditions. --JakubNarebski (talk) 21:18, 3 August 2018 (UTC)

It's a false dilemma. Complex numbers are vectors ($\mathbb{C}$ is a two-dimensional $\mathbb{R}$-vector space, and more generally every field is a vector space over any subfield), but that doesn't change anything about the fact that $i$ is by definition a square root of -1. Zmatt (talk) 20:38, 3 August 2018 (UTC)

Fun factoid: not only is $\mathbb{C}$ the unique proper field extension of finite degree over $\mathbb{R}$ (since $\mathbb{C}$ is algebraically closed), but the converse is true as well: $\mathbb{R}$ is the only proper subfield of finite index in $\mathbb{C}$. They're like a weird married couple. Zmatt (talk) 20:53, 3 August 2018 (UTC)

Altho there are no "meta-abelian" groups there are metabelian groups. If xy=yx then the commutator [x,y]=xyx^{-1}y^{-1}=1. The group generated by the commutators -- the commutator subgroup -- is thus a measure of how far a group is from being abelian. A metabelian group is a nonabelian group whose commutator subgroup is abelian. Thus a metabelian group is one made of a stack of two abelian groups. It is "meta-abelian" in that sense. A standard example is the group of invertible upper-trianglular matrices. The commutators all have 1s on the diagonals.