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?.

> "and behave as the corresponding operations on rational and real numbers do."

Here it may be better to say

> and behave **in certain specified respects** as the corresponding operations on rational and real numbers do.

And of course a textbook definition will be explicit about those specified respects.

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 $0.$