I’ve been putting off revising the *Rings & Arithmetic* course since I
didn’t follow the lectures or really do the problems back in Michaelmas,
but people keep telling me it’s nice and actually quite easy, so I’m
finally doing some—it’s a compulsory course, but I only have to answer
one question so I thought I might just skip it and try and put more time
into the others. I think that having done some Number Theory helps a
great deal. Fellow Hungarian mathematician who did most of the second
year number theory course at school says this helped a lot; they did
this while we were wasting our time with calculus.

So I’m starting to like algebra more—especially now that quotient spaces
don’t scare me quite so much—but this is the usual thing that I really
like Maths when I can do it, but I can’t do it often enough in this
degree course that I stop enjoying it. I wish I’d had a better year so I
could have a better grasp of courses like this; I should have persevered
with Fields, too. Also for any maths undergrads reading, *Abstract
Algebra* by I.N. Herstein (Chichester: Wiley 1999) is definitely worth
checking out; doesn’t quite do enough for my second year rings course,
but shows you what’s important about the topics.

Here’s the coolest thing I’ve found so far, where we use quotients of
rings to construct ℂ, rather than just saying that there’s an element
*i* that squares to − 1, which appeals to my constructivist leanings.

We have the ring of polynomials with real coefficients, ℝ[*X*], no
problem, and now write ⟨*X*^{2} + 1⟩ for the multiples of the
polynomial *X*^{2} + 1.[1] It’s easy to show that $\langle
X^{2}+1\rangle \vartriangleleft \mathbb{R}[X]$. So we can form the
quotient ring ℝ[*X*]/⟨*X*^{2} + 1⟩, and this is essentially ℂ.
So what’s our imaginary unit? Well write
$\mathrm{i} \mathrel{\mathop:}= \overline{X} \mathrel{\mathop:}= X +
\langle X^{2}+1\rangle$ and then using the division algorithm we can get
that every element can be written *a* + *b**i*: every polynomial may be
written as *a* + *b**X* + *g*(*X*)(*X*^{2} + 1)). And indeed

$$
\mathrm{i}^2 = \overline{X}^2 = \overline{X^{2}} = \overline{X^{2} + 1 -
1} = \overline{X^{2} + 1} - \overline{1} = 0 - 1 = -1
$$

as we require.

[1] The reason we use *X* rather than *x* here is to emphasise that
polynomials are *not* functions, just formal expressions of coefficients
and a few *X*. *x* would traditionally denote a variable in a function
rather than just a component of an expression.

It’s strange: for me, this sort of construction of i is just the same, really as the “standard” “this is a number which squares to -1” non-construction. Feels just as arbitrary or not.