Field

Definition 8. A field \(\mathbb{F}\) is a ring where the set \(\mathbb{F} \setminus \set{0}\) is an Abelian group under multiplication with the multiplicative identity 1.

Remark bz. Fields with a finite number of elements are finite fields and are often denoted \(\mathbb{F}_q\) or \(\mathrm{GF}(q)\), where \(q\) is the order of the field. Since every field is a commutative ring, the only difference between rings and fields is that in a field, every element other than \(0\) has its multiplicative inverse.

Remark ba. The finite field \(\mathbb{Z}_2\) is denoted \(\mathbb{F}_2\) or \(\mathrm{GF}(2)\). The additive and multiplicative identities are \(0\) and \(1\), respectively. The element \(1\) is its own additive and multiplicative inverse. Addition in this field corresponds to the exclusive or operation denoted XOR or \(\oplus\), and subtraction is identical to addition.

Example dk.

The sets \(\mathbb{Q}\), \(\mathbb{R}\), and \(\mathbb{C}\) are fields with their standard addition and multiplication.
The integers do not form a field since not all elements have their multiplicative inverse in this set.
The set of integers modulo \(p \in \mathbb{Z}\), denoted \(\mathbb{Z}_p\), is a field whenever \(p\) is prime. The primality of \(p\) ensures that each non-zero element has its multiplicative inverse, which can be found using the extended Euclidean algorithm.

Definition b3. If \(\mathbb{F}\) is a subfield of a field \(\mathbb{E}\), then \(\mathbb{E}\) is an extension field of \(\mathbb{F}\).

For example, \(\mathrm{GF}(2^2)\) is an extension field of \(\mathrm{GF}(2)\). A more detailed way of constructing extension fields is given in the proof of Proposition na.