I.3 More Algebraic Structures: Fields, Vector Spaces, Rings and Creating New Strucures out of Old Ones: Substructures and Products, Pages 20 – 24.

A field has two binary operations.  The operations are both commutative, associative and both have identity elements.  Each eleemnt has an inverse except for the identity element of one of the operations (zero if the operations are addition and multiplication).  Also, the distributative law must apply, this ties the two operations together.  If the operations are addition and multiplication then the distributative law is x * (y + z) = x * y + x * z.

The number systems Q, R and C form fields with addition and multiplication.  Also, Fp = the set of integers modulo a prime p, with addition and multiplication also defined modulo p form a field.

One can create a new field by adjoining the roots of polynomials which are not already in F to a field F creating a new extended field F’.  This was done to create the complex numbers from the reals with the roots of the polynomial P(x) =  x2 + 1.  The exact same process can be used with, for example, F5 as the field and appending i to the elements.  We can also append roots to the rationals forming the extended field Q(Ö2) or in general Q(g), where g Î root of a polynomial which is not a rational number.   

Filed extensions can be interesting and are used in automorphisms and vector spaces. 

Vector spaces are formed as linear combinations of some basis vectors.  Given scalers a and b and vectors v and w the element av + bw is also in the vector space.  Addition of vectors must be associative and commutative with an identity and inverse element for each v under addition.  Scaler multiplication must be associative a(bv) = ab(v).  Two distributative laws must be satisfied: (a + b)v = av +bv and a(v + w) = av + aw for any scalers a and b and any vectors v and w.   

A set of vectors forms a basis for V if every vector in V can be written in exactly one way as a linear combination of the set of basis vectors.  A set of vectors spans the space V if there is at least one linear combination of the spanning set that equals each vector in V.  If there is a spanning set where no vector may be formed by more than one linear combination of spanning set then the spanning set vectors are said to be independent.

The number of elements in a basis of V is said to be the dimension of V. This number is unique.  A vector space may have an infinite number of basis vectors, then the vector space is called infinite dimensional. 

The vectors in a vector space may be functions.  The scalars used to define a vector space may be from any field, F.  In this case one is said to form a vector space V over F.

Notice again that a vector space over Cartesian coordinates do not occur in the actual world because there is not Euclidean geometry.

Another algebraic structure is a ring.  A ring has most but not necessarily all of the properties of a field.  Requirements of the multiplication operation are less stringent.  For example, non zero elements of the ring are not required to have multiplicative inverses.  Sometimes the multiplication is not required to be commutative.  Although there are commutative rings.  Examples of commutative rings are the set of integers and the set of all polynomials with coefficients in some field F. 

Structures that we already know may be used to create new structures.  These new structures are important to use as examples and counterexamples.  Examples are important in mathematical thought. 

One possible group is the group G of all symmetries of an icosohedron.

Substructures may also be important.  For example, primes are a subset of the integers and the set a + b i with a and b rational are an interesting subset of the complex numbers.

In general groups have subgroups, vector spaces have subspaces and rings have subrings.  Even the plane has interesting subsets such as the Mandelbrot set.   

Products of groups may be defined.  Let G and H be two groups then the product group G x H.  Using the binary operation from these groups we may form (g1,h1) (g2,h2) = (g1g2,h1h2).  The binary operation from G is used on the first coordinate and that from H is used on the second coordinate.

Products of vector spaces V and W, or V x W, are all pairs (v,w) with v in V and w in W.  Addition and scaler multiplication are defined as  (v1,w1) + (v2,w2) = (v1 + v2 , w1 + w2) and l(v,w) = (lv,lw).   The dimension of the resulting space is the sum of the dimensions of V and W.  The space V x W is usually written V Å W.  It is called the direct sum of V and W.

It is not always possible to define products structures that retain the necesary properties in this way.  Sometimes we can change the multiplication definition to satisfy the necessary properties.

We can define more complicated products of groups than the direct product defined above.  Consider the dihedral group D4 which is the group of all symmetries of a square.  We can create this group as comprised of 90 degree turns and reflections.  D4 is the product of the group {I, T1, T2, T3}, consisting of four rotations or turns and {I,R} consisting of the identity and a reflection.  One must be careful to define multiplication to enforce RTR = T-1.  This is an example of a semi-direct product of two groups.  Semi-direct groups may be formed by defining different binary operations of the set of pairs (g,h).

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