Field (mathematics)

In mathematics, field theory usually refers to a subdiscipline of abstract algebra that studies fields, which are mathematical constructs that generalize on the familiar concepts of real number arithmetic.

The terms field and field theory carry other meanings in other areas of mathematics, notably in calculus and mathematical physics.

This article deals exclusively with fields and field theory as the terms are used in abstract algebra.

Informal description and definition

When dealing with $\mathbb {Q}$, the set of rational numbers, we notice several things:

• The rational numbers form what mathematicians call a commutative group under addition.
• When we exclude the number 0, they form a group under multiplication as well.
• For any triplet $a,b,c$, we have that $a(b+c)=ab+ac$ - both calculations yield the same answer.

Dealing with other sets, both finite and infinite, we often notice this behavior. Obvious examples are $\mathbb {R}$, the set of real numbers and $\mathbb {C}$, the set of complex numbers.

Less obvious examples are Z₂, the set {0,1} under addition and multiplication modulo 2. Other examples are Z₃ and Z₅, ... , Z_p. In general, any set {0,1,...,p-1} under addition and multiplication modulo p, for any prime p, is a field.

All these sets, and others where the three conditions listed above are fullfilled, are known as fields by mathematicians.

It's important to notice that neither the field elements nor the binary operations necessarily have to be anything closely resembling numbers. It's sufficient that the actual system under study can be conceptualized into a structure that satisfies the formal definition of a field:

Formal definition of a field

A field consists of a set F, along with a binary operation + on F such that F is a commutative group with an identity element 0; and another binary operation * on F such that F\{0} is a commutative group. Also, for any $a,b,c\in F$, we must have that $a*(b+c)=a*b+a*c$.

The first binary operation is usually called addition and the second one multiplication.

Characteristic of a field

Characteristic of a field is a such natural number $k$ that $0=1+1+\cdots +1$ ($k$ ones in the right part). If such natural number doesn't exist, the characteristic of a field take to be 0. If the characteristic of a field is non zero, it is a prime number because non-coprime with $k$ numbers would not have reverse element otherwise.

See also

• Skew-field
• Algebra
• Vector space

Related topics

• Galois theory
• Field extension

External links

• Field at MathWorld