Vector spaces

Field

Field

A Field \(F\) is a commutative division ring, that is, it is a commutative ring and every elements except \(0\) in \(F\) has a multiplicative inverse in \(F\).

What is a field?

1. Is \(\mathbb{Z}\), the set of all integers, a field?
2. Is \(\mathbb{Z}/2\mathbb{Z}\) a field? Find a sufficient condition for \(\mathbb{Z}/m\mathbb{Z}\) to be a field, where \(m\in\mathbb{N},m\ge2\).

Ans

1. No. \(2\) has no multiplicative inverse in \(\mathbb{Z}\).
2. Yes. If \(m\) is a prime, then it is a field by the Lagrange theorem of subgroups.