Appendix A

Summary

Fields

Fields are important algebraic structures used in almost all branches of mathematics. Here we only cover the definitions and theorems needed for the purposes of this book.

Definition. A field F is a set along with two operations (denoted with addition and multiplication notation) on pairs of elements of F such that the following properties are satisfied.

1. For all a and b in F, we have that a + b ∈ F.

2. For all a, b, and c in F, we have that (a + b) + c = a + (b + c).

3. There exists an element 0 in F satisfying a + 0 = a for all a ∈ F.

4. For every a ∈ F there exists a b in F such that a + b = 0.

5. For all a and b in F we have that a + b = b + a.

6. For all a and b in F we have that ab ∈ F.

7. For all a, b, and c in F we have that (ab)c = a(bc).

8. There is an element 1 in F satisfying 1a = a for all a ∈ F.

9. For every a ∈ F with a ≠ = 0, there exists a b ∈ F such that ab = 1.

10. For every a and b in F we have that ab = ba.

11. For every a, b, and c in F we have that a(b + c) = ab + ac.