Abstract algebra

Abstract algebra is my favourite area of mathematics. It is the study of abstract algebraic structures, most notably groups, rings and fields.

A group is a pair (G,·) consisting of a non-empty set G and a binary operation · on G such that · is associative in G, there is an identity element in G with respect to · (i.e. an element e ∈ G such that g·e = e·g = g for all g ∈ G) and every element in G has an inverse element in G with respect to · (i.e. for all g ∈ G there exists g⁻¹ ∈ G such that g·g⁻¹ = g⁻¹·g = e). (Note: the identity element of a group is unique, and given any element in the group its inverse is unique.) Often, when the binary operation is understood, the set G itself (rather than the pair (G,·)) is called a group. Note that while · has to be associative, it need not be commutative; if it is, then G is said to be an Abelian group under ·.

A ring is a triple (R,+,×) consisting of a non-empty set R and two binary operations on R called addition + and multiplication × such that (R,+) is an Abelian group, × is associative in R and distributive over + (i.e. for all a, b, c ∈ R, a×(b+c) = (a×b)+(a×c) and (b+c)×a = (b×a)+(c×a)). The additive identity is denoted 0_R; many authors also insist that a ring must have a multiplicative identity, 1_R. Again, if the operations + and × are clear from the context, the set R itself, rather than the cumbersome triple (R,+,×), is referred to as a ring.

A field is a ring (F,+,×) (as defined above) such that (F\{0_F},×) is an Abelian group. And, once again, it is usual to call just F itself a field if + and × are understood.

In 2001, I celebrated my love for abstract algebra by composing an English sonnet about groups, rings and fields.

Lagrange’s theorem states for finite G
That every subgroup’s order must divide
G’s – if G’s prime then simple must G be.
All finite simple groups are classified.
Let R be any ring. All subrings I
Closed under multiplying are ideals.
One can construct then quotient rings R by
I – good for (a subset of the reals).
Real numbers form a field – and this consists
Of quite a structure one can marvel at.
Why, given each prime power, there exists
A Galois field of order simply that.
Nice is the study of groups, rings, and fields –
Great satisfaction to the mind it yields.

And representation theory is a combination of group theory and linear algebra!

During my first year at Queen Mary and Westfield College – now Queen Mary, University of London – my course adviser was Professor Stephen Donkin. (He was actually only Doctor Donkin when my course started – but was very soon promoted to Professor Donkin.) And his area of specialism was representation theory. Indeed, the theory of Galois representations holds pride of place in the history of mathematics, being one of the tools employed by Andrew Wiles in his groundbreaking proof of Fermat’s last theorem – indirectly by proving the special case of the Taniyama–Shimura conjecture for semistable elliptic curves over (in particular the Frey elliptic curve).

Let us look more closely at the complex transformation

where the coefficients are integers with ad − bc = 1. There are two cases to consider: c = 0, and c ≠ 0.

Case 1: c = 0
In this case, due to the condition ad − bc = 1, we simply get the linear translations f(z) = z ± b for all z ∈ .

Case 2: c ≠ 0
Then f(z) is defined for all z ∈ except z = −d/c. Also

implies

which is not defined for w = a/c. To get round these little obstacles, we simply set f(−d/c) = ∞ and f(∞) = a/c! Mathematicians need to be clever sometimes.

The extension process has the unfortunate effect of removing certain algebraic properties from , especially the fact that the extended complex plane is no longer a field (∞ does not have an additive or multiplicative inverse). But the addition of a point at infinity has the advantage of enriching certain topological properties of the complex plane, especially the fact that the extended complex plane is compact (whereas is not). Indeed is homeomorphic to the surface of a sphere; it is also called the Riemann sphere.

But this is steering the thread away from the main topic, which is abstract algebra, to the realms of topology and projective geometry. Let us return to the main topic – though I would not mind delving into algebraic topology, particularly homotopy theory, myself.