A short primer on algebra

[Eric Hughes]

In the interest of good mathematical terminology, here is a short
primer on the most basic algebraic structures.  The definitions are
not complete but rather evocative and are designed to prevent
confusion.

Field -- has addition, subtraction, multiplication, and division.
Examples are the real numbers (R), the complex numbers (C), and the
rational numbers (Q).  An important class of fields for crypto are
integers modulo a prime (Z/pZ or F_p).  An important class of fields
for error coding are polynomials with binary coeffients modulo an
irreducible polynomial (F_2[x]/p(x)F_2[x]).

Ring -- has addition, subtraction, multiplication, but no division.
Every field is a ring but not vice-versa.  Examples are the integers
(Z), the integers modulo a composite number (Z/nZ) and polynomials
with various rings, including R[x], Z[x].

Group -- has either addition/subtraction or multiplication/division,
but not necessarily both.  Every ring is a group under addition, but
not vice-versa.  If the group is commutative, we write the operation
as addition typically; if not, we use multiplication.  Examples of
commutative groups are solutions of an elliptic curves and rotations
in the plane.  Examples of non-commutative groups are permutations,
rotations in three dimensions, and Euclidean transformations of the
plane.

Eric