Field

Def

Before we examine a rigorous definition of a field, lets cover some examples

$\mathbb{Q}$ is a field however $\mathbb{Z}$ is not a field, lets inspect the important difference between the two sets

1. They both have closure under addition :
   • take any two numbers integers, say for sake of example, 2 and 3, 2 + 3 = 5 which is also an integer. a more mathematical way of stating this statement would be let $2,3\in \mathbb{Z}$, since $2+3=5$ and $5\in \mathbb{Z}$, then $2+3\in \mathbb{Z}$
   • Similar with rational numbers, take $\frac{1}{2},\frac{1}{3}\in \mathbb{Q}$, $\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\in \mathbb{Q}$
2. They both have closure under multiplication
3. They both have additive inverses

However, the two sets differ as one has multiplicative inverses while the other does not. $\mathbb{Z}$ does not have any multiplicative inverses, which makes it fail to be a field!

Challenge

Prove, for any two rational numbers $a,b\in \mathbb{Q}$, $a+b\in \mathbb{Q}$

Field

A field is a set F(also denoted as a set K) with two binary operations : + : F \times F \to F$$$$\cdot : F \times F \to F such that it is an abelian ring under both operations and it satisfies the following axioms:

Field Axioms