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).
A field is a commutative ring with unity.
An integral domain is a commutative ring with unity and no zero-divisors.
A zero-divisor is a non-zero element 'a' of a commutative ring 'R' such that there is a non-zero element bR with ab=0
|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
Examples of Fields
- 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).
Every field is an integral domain
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.
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.
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.
|splitting of p(x)|
|x³-2||Q||Q(w,³√2)||(x-³√2)(x-w ³√2)(x-w² ³√2)|
|Q(w)[z]/<z³-2>||(x-z)(x-w z)(x-w² z)|
|Q(³√2)[y]/<y² + y + 1>||(x-³√2)(x-y ³√2)(x-y² ³√2)|