#math/linear-algebra #flashcards/math A **Field** is a set with the following requirements: 1. Contains at least two distinct elements 0 and 1. 2. Has the `operations` of addition and multiplication. 3. Is commutative; $AB=BA.$ 4. Is associative; $A+(B+C)=(A+B)+C$. 5. Maintains additive identity; $A+0=A$. 6. Maintains multiplicative identity; $A\times1=A$ 7. Every element $A$, has a unique additive inverse; $A+(-A)=0$. 8. Every element $A$, besides 0, has a unique multiplicative inverse; $AB=1$. 9. Is distributive; $A(B+C)=AB+AC$. We use $\mathbf{F} \text{ to denote } \mathbb{C} \text{ or } \mathbb{R},$ both of which are fields whose elements are scalars. --- `Operation`: A function $\mathbf{F}\times \mathbf{F}\to \mathbf{F}$ that takes two field elements and returns another field element. Addition and multiplication are operations.