1. The problem is to understand the definition and properties of a finite field, focusing on the addition operation. 2. A field $F$ is a non-empty set with two operations: addition (+) and multiplication (\cdot), satisfying certain properties. 3. For addition, the properties are: (1) Closure: For any $a, b \in F$, the sum $a + b$ is also in $F$. (2) Commutativity: For any $a, b \in F$, $a + b = b + a$. (3) Associativity: For any $a, b, c \in F$, $(a + b) + c = a + (b + c)$. (4) Additive Identity: There exists an element $0 \in F$ such that for any $a \in F$, $a + 0 = a = 0 + a$. 4. These properties ensure that addition in $F$ behaves like familiar addition in numbers, making $F$ an algebraic structure suitable for further operations. 5. Understanding these properties is fundamental to working with finite fields in algebra and applications like coding theory and cryptography.