Field

Dependencies:

1. Group
2. Ring

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:

1. Zp is a field
2. A field is an integral domain
3. I is a maximal ideal iff R/I is a field
4. A finite integral domain is a field
5. Being a field is preserved under isomorphism
6. Elementary row operation is matrix pre-multiplication
7. Rank of a matrix
8. Inverse of a matrix
9. Every elementary row operation has a unique inverse
10. Row equivalent matrices have the same row space
11. Row equivalence of matrices
12. A matrix is full-rank iff its determinant is non-0
13. Vector Space
14. Basis of F^n
15. System of linear equations
16. Inner product space
17. Polynomial GCD theorem
18. A polynomial of degree n has at most n zeros
19. Polynomial division theorem

Info:

• Depth: 2
• Number of transitive dependencies: 2

Transitive dependencies:

1. Group
2. Ring