# Field

## Dependencies:

The ring $(R, +, \circ)$ is a field iff all of the following conditions are met:

- $(R, +)$ is a commutative group (identity is denoted as 0).
- $(R - \{0\}, \circ)$ is a commutative group (identity is denoted as 1).
- Distributive property: $\forall a, b, c \in R, (a(b+c) = ab+ac \wedge (a+b)c = ac+bc)$.

Consequently, a field is also a commutative ring.

Also, a field has at least 2 elements, 0 and 1.

Common examples of fields are $(\mathbb{Q}, +, \times)$, $(\mathbb{R}, +, \times)$ and $(\mathbb{C}, +, \times)$ (it is trivial to prove that they are fields).

## Dependency for:

- I is a maximal ideal iff R/I is a field
- Zp is a field
- A field is an integral domain
- A finite integral domain is a field
- Being a field is preserved under isomorphism
- Inner product space
- Vector Space
- Basis of F^n
- Row equivalence of matrices
- Inverse of a matrix
- Rank of a matrix
- Row equivalent matrices have the same row space
- Elementary row operation is matrix pre-multiplication
- Every elementary row operation has a unique inverse
- A matrix is full-rank iff its determinant is non-0
- System of linear equations
- Polynomial GCD theorem
- Polynomial division theorem
- A polynomial of degree n has at most n zeros

## Info:

- Depth: 2
- Number of transitive dependencies: 2