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Abstract Algebra

- a brief survey by Bengt Månsson

 

The elements of abstract algebra may look like a heap of definitions, and when it starts to become really interesting it also gets quite difficult. This page is just a start. Nevertheless I hope you will understand how earlier known results get a common ground. This page will also provide some necessary background for coming surveys on Galois Theory and Algebraic Number Theory. My intention is to make at least some essential part of these important and interesting areas of mathematics understandable to interested students at senior high school level or slightly above.

In examples we will use the standard notations Z, Q, R, C for the set of integers, rational numbers, real numbers, and complex numbers, respectively.

 

Binary Operations

or composition rules are e g addition and multiplication. We will make this precise and more general.
   
A binary operation on a non-empty set M is a function from M×M to M.
 

Groups

 
A group is an algebraic structure (G,*) such that 
    (i)   * is associative, 

    (ii)   there is a neutral element e in G such that a*e = e*a = a for all a in G

    (iii)  for each a in G there is an inverse a' in G such that a*a' = a'*a = e.

 

Rings and Fields

 
A ring is an algebraic structure (R,+,·) such that 
    (i)   (R,+) is an abelian group, 

    (ii)   · is associative, 

    (iii)  a·(b + c) = a·b + a·c  and  (a + bc = a·c + b·c  for all a, b, c in R.

A field is a ring (K,+,·) such that 
    (iv)  (K - {0},·)  is an abelian group.
 

Homomorphism and Isomorphism

 
A group homomorphism f is a function from a group G to a group G' such that 
    f(ab) = f(a)f(b)  for all a, b in G.
A ring homomorphism f is a function from a ring R to a ring R' such that 
    (i)   f(a + b) = f(a) + f(b)  for all a, b in R

    (ii)   f(a·b) = f(af(b)  for all a, b in R.

   

Normal Subgroups and Quotient Groups

Ideals and Quotient Rings

Field Extensions

 
A simple algebraic extension is an extension K(a):K such that a is a zero 
of a non-zero polynomial over K. a is then said to be algebraic over K.
A simple transcendental extension is an extension K(a):K where a is not 
a zero of any non-zero polynomial over K. a is then said to be 
transcendental over K.
   
If K(a):K is a simple algebraic extension, then there is a unique, 
monic polynomial m(t) over K of lowest degree, such that m(a) = 0. 
The polynomial m(t) is irreducible, and divides every polynomial 
over K, of which a is a zero.
   
If m(t) is a monic, irreducible polynomial over the field K, then there is 
an extension K(a):K, such that a has minimum polynomial m(t) over K.
 

Vector Spaces

 
V is a vector space over a field F if (V,+) is an abelian group, and there is 
defined a multiplication of elements in V with elements in F, such that 
    (i)   a(u + v) = au + av  for all a in F and u, v in V

    (ii)  (a + b)u = au + bu  for all a, b in F and u in V

    (iii)  a(bu) = (ab)u  for all a, b in F and u in V

    (iv)  1u = u  for all u in V

    (v)   0u = 0  for all u in V.

   
If K(a):K is a simple algebraic extension, then [K(a):K] equals 
the degree of the minimum polynomial of a over K.
 

Conclusion


© Copyright Bengt Månsson 1997, bengtmn@algonet.se. Last updated November 11, 1997.