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Field with four elements

The smallest non-prime field is the field with four elements, which is commonly denoted GF(4) or F 4. {\displaystyle \mathbb {F} _{4}.} It consists of the four elements 0, 1, α, 1 + α such that α = 1 + α, 1 ⋅ α = α ⋅ 1 = α, x + x = 0, and x ⋅ 0 = 0 ⋅ x = 0, for every x ∈ GF(4), the other operation results being easily deduced from the distributive law. See below for the complete operation tables.

This may be deduced as follows from the results of the preceding section.

Over GF(2), there is only one irreducible polynomial of degree 2: X 2 + X + 1 {\displaystyle X^{2}+X+1} Therefore, for GF(4) the construction of the preceding section must involve this polynomial, and G F (4) = G F (2) / (X 2 + X + 1). {\displaystyle \mathrm {GF} (4)=\mathrm {GF} (2)/(X^{2}+X+1).}

Let α denote a root of this polynomial in GF(4). This implies that

and that α and 1 + α are the elements of GF(4) that are not in GF(2). The tables of the operations in GF(4) result from this, and are as follows:

A table for subtraction is not given, because subtraction is identical to addition, as is the case for every field of characteristic 2.In the third table, for the division of x by y, the values of x must be read in the left column, and the values of y in the top row. (Because 0 ⋅ z = 0 for every z in every ring the division by 0 has to remain undefined.) From the tables, it can be seen that the additive structure of GF(4) is isomorphic to the Klein four-group, while the non-zero multiplicative structure is isomorphic to the group Z.

The map φ: x ↦ x 2 {\displaystyle \varphi:x\mapsto x^{2}} is the non-trivial field automorphism, called the Frobenius automorphism, which sends α into the second root 1 + α of the above mentioned irreducible polynomial X + X + 1.