note: The concept of a 'field' in algebra is different from the concept of a 'vector or scalar field' which occurs in geometry and physics. (although there is a connection with vectors as we shall see).
Definitions
A field is a commutative ring with unity.
An integral domain is a commutative ring with unity and no zerodivisors.
A zerodivisor is a nonzero element 'a' of a commutative ring 'R' such that there is a nonzero element bR with ab=0
Axioms
axiom  addition  multiplication  both 

associativity  (a+b)+c=a+(b+c)  (a*b)*c=a*(b*c)  
commutativity  a+b=b+a  a*b=b*a  
distributivity  a*(b+c)=a*b+a*c (a+b)*c=a*c+b*c 

identity  a+0 = a 0+a = a 
a*1 = a 1*a = a 

inverses  a+(a) = 0 (a)+a = 0 
a*a^{1 }= 1 a^{1}*a = 1 if a≠0 
Examples of Fields
Examples include:
 the real number field.
 the complex numbers C.
 the rational numbers Q.
 finite fields.
 various fields of functions.
Some algebras that are not fields are vectors (unless a commutative multiplication is used), matrices (not commutative) and quaternions (not commutative).
Properties
Every field is an integral domain
Extension Field
A Field 'E' is an extension field of a field 'F' if F is a subset of E and the operations of F are those of E restricted to F.
Polynomial Domain
Every algebraic number field can be obtained as the quotient of the polynomial domain Q[x] by the principle ideal generated by an irreducible polynomial.
Theorems
The degree of an extension Q(r) always matches the degree of the irreducible polynomial to which r is a root.
The degree of a normal extension matches the degree of its Galois group.
p(x)  F Field 
E Extension 
splitting of p(x) 

x²2  Q  Q(√2)  (x√2)(x+√2) 
Q[z]/<z²2>  (xz)(x+z)  
x³2  Q  Q(w,³√2)  (x³√2)(xw ³√2)(xw² ³√2) 
Q(w)[z]/<z³2>  (xz)(xw z)(xw² z)  
Q(³√2)[y]/<y² + y + 1>  (x³√2)(xy ³√2)(xy² ³√2)  