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 finiteG

That every subgroup’s order must divideG’s – ifG’s prime then simple mustGbe.

All finite simple groups are classified.

LetRbe any ring. All subringsI

Closed under multiplying are ideals.

One can construct then quotient ringsRbyI– 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.