# Maths - Abstract Algebra and Discrete Mathematics

### Abstraction

One of the most important approaches is abstraction. In mathematics we may look at specific structures and then 'abstract' away the details to get to more general results.

For example, if we discover a structure, we try to discover the laws which govern it (usually in the form of equations - identities or axioms). We can then consider these laws on their own and work out the most general structures which satisfy them.

### Structures

This section contains maths topics which involve abstract concepts and discrete (non continuous) mathematics. ### Terminology

• Field - Commutative division ring (different meaning of the word 'field' than fields on a manifold)
• Integral Domain - Commutative Ring with unity and no zero divisors
• Vector Space - Vectors over a field, Addition with scalar multiplication (vector & field (scalars))
• a(v+u) = a v + a u
• (a + b) v = a v + b v
• a (b v) = (a b) v
• 1 v = v
• Zero Divisor - A non-zero element 'a' of a commutative ring 'R' such that there is a non-zero element b R with ab=0

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.      The Princeton Companion to Mathematics - This is a big book that attempts to give a wide overview of the whole of mathematics, inevitably there are many things missing, but it gives a good insight into the history, concepts, branches, theorems and wider perspective of mathematics. It is well written and, if you are interested in maths, this is the type of book where you can open a page at random and find something interesting to read. To some extent it can be used as a reference book, although it doesn't have tables of formula for trig functions and so on, but where it is most useful is when you want to read about various topics to find out which topics are interesting and relevant to you.