# Definition of Field in mathematics

Wikipedia definition: In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do.

My question is regarding closure. Does the word defined entail closed?. Is a field by definition closed under all these operations?.

• Yes, the operations are (apart from division because of obvious reasons) $\mathbb{K} \times \mathbb{K} \to \mathbb{K}$. – LucaMac Aug 27 '18 at 1:10
• Yes, the closure axiom is implicit. – Jair Taylor Aug 27 '18 at 1:30
• If you go to the "definition" section of the Wikipedia article, you'll get answers to all your questions. – enedil Aug 27 '18 at 1:31

Notice that with the the usual operations on $\mathbb Z = \{0, \pm1, \pm2, \pm3, \ldots\}$ you can divide in some cases, e.g. $1333 \div 31 = 43,$ but $\mathbb Z$ is not closed under division since in most cases you cannot divide while remaning within $\mathbb Z.$ For example $9$ is not divisible by $2.$ Thus $\mathbb Z$ is not a field.
Closure under those operations holds in field, except that one cannot divide by $0.$