 # 2. Fields

### Definition

A set is a field if it is equipped with two two-argument operations and called addition and multiplication and two selected elements , satisfying the following conditions for every :

1. addition is associative: ,
2. multiplication is associative: ,
3. addition is commutative: ,
4. multiplication is commutative: ,
5. is the neutral element for addition: ,
6. is the neutral element for multiplication: ,
7. there are negations: there exists , such that , such is denoted by ,
8. there are reciprocals: if , there exists , such that , such is denoted by or ,
9. these operations are distributive: .

It is worth noticing that there exists exactly one negative for i and exactly one reciprocal for (if ). Indeed, if , then .

### Examples

Examples: real numbers with standard operations and standard and , similarly rational numbers .

Those examples are infinite fields, but there also exist finite fields. The simplest example of a finite field is , for a prime number . , and addition and multiplications are taken modulo . E.g. in the operation work as follows:  ### Subfields is a subfield of a field , if for every , , and for , , and moreover . Obviously a subfield of a field is a field.

E.g., is a subfield of .

### Systems of linear equations over a given field

We can also solve systems of linear equations with coefficients and variables in a given field. We do this using the same method which was described for real numbers, i.e. transforming a matrix into reduced echelon form, but obviously calculating everything in the given field. E.g. to solve in the system we get:    So the general solution takes the following form: which gives . Substituting, for example, , we get and exemplary solution — and it is easy to check that those numbers satisfy the equations from the system (again in ).

### Extensions by an element

Given a subfield of a field and an element , we will denote by the least field containing . Notice that if is a root of a polynomial , then (indeed we can assume that is not a root of , and then for some , so , and , so ). In particular, if , then .