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## Tuesday, November 10, 2009

### Watched lecture 8 of Abstract Algebra E-222

The following topics are discussed in this lecture:

Isomorphism Theorem

Vector spaces over an arbitrary field
- Definition field
- Examples of finite fields

Proof that Z/pZ is a field
- Added to what we know from Z/nZ as an additive subgroup of Z we must prove that each a in Z/pZ has a multiplicative inverse, so we must show that if a is not a multiple of p then there is an integer b such that a*b congruent 1 mod p.
( Actual proof is worked out in the video )

What are the finite fields beyond Z/pZ ?
- the finite fields are of order p^n where p is a prime and n>=1, so there are finite fields of order 2, 3, 4, 5, 7, 8, 9, 11, etc. ( note 6 = 2*3, 10=2*5 not of type p^n )

Definition of a vector space ( V )
- With a map f: VxF -> V which is called the scalar multiplication
- ( All rules are written down on board. )

Examples of vector spaces
- V={0}
- V=F
- V=F2
- V=Fn
- V=F[X], vector space of all polynomials p(x) with coefficients in F

Vector subspace
- A subgroup 'stable' under scalar multiplication

Vector space homomorphisms
- Linear transformations ( as we knew it ) are explained as group homomorphisms stable under scalar multiplication
- So for T: V-> W we can define the Kernel of T as a subspace of V and the image of T as a subspace of W. We can also define the quotient space V/W analog to the quotient group

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Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)