Editing Finite field (section)

=== Field with four elements ===
The smallest non-prime field is the field with four elements, which is commonly denoted <math>\mathrm{GF}(4)</math> or <math>\mathbb{F}_4.</math> It consists of the four elements <math>0, 1, \alpha, 1 + \alpha</math> such that <math>\alpha^2=1+\alpha</math>, <math>1 \cdot \alpha = \alpha \cdot 1 = \alpha</math>, <math>x+x=0</math>, and <math>x \cdot 0 = 0 \cdot x = 0</math>, for every <math>x \in \mathrm{GF}(4)</math>, 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 <math>\mathrm{GF}(2)</math>, there is only one [[irreducible polynomial]] of degree <math>2</math>:
<math display="block">X^2+X+1</math>
Therefore, for <math>\mathrm{GF}(4)</math> the construction of the preceding section must involve this polynomial, and
<math display="block">\mathrm{GF}(4) = \mathrm{GF}(2)[X]/(X^2+X+1).</math>
Let <math>\alpha</math> denote a root of this polynomial in <math>\mathrm{GF}(4)</math>. This implies that
<math display="block">\alpha^2 = 1 + \alpha,</math>
and that <math>\alpha</math> and <math>1+\alpha</math> are the elements of <math>\mathrm{GF}(4)</math> that are not in <math>\mathrm{GF}(2)</math>. The tables of the operations in <math>\mathrm{GF}(4)</math> result from this, and are as follows:
{| style="border-spacing:1.7em 0"
|
{| class="wikitable"
|+ Addition <math>x + y</math>
|-
! style="width:24%;" {{diagonal split header|{{math|''x''}}|{{math|''y''}}}} !! style="width:20%;"| {{math|0}} !! style="width:20%;"| {{math|1}} !! style="width:20%;"| {{math|''α''}} !! style="width:20%;"| {{math|1 + ''α''}}
|-
!style="text-align:left"| {{math|0}}
| {{math|0}}
| {{math|1}}
| {{math|''α''}}
| {{math|1 + ''α''}}
|-
!style="text-align:left"| {{math|1}}
| {{math|1}}
| {{math|0}}
| {{math|1 + ''α''}}
| {{math|''α''}}
|-
!style="text-align:left"| {{math|''α''}}
|| {{math|''α''}}
| {{math|1 + ''α''}}
| {{math|0}}
| {{math|1}}
|-
!style="text-align:left"| {{math|1 + ''α''}}
| {{math|1 + ''α''}}
| {{math|''α''}}
| {{math|1}}
| {{math|0}}
|}
|
{| class="wikitable"
|+ Multiplication <math>x \cdot y</math>
|-
! style="width:28%;" {{diagonal split header|{{math|''x''}}|{{math|''y''}}}} !! style="width:12%;"| {{math|0}} !! style="width:20%;"| {{math|1}} !! style="width:20%;"| {{math|''α''}} !! style="width:20%;"| {{math|1 + ''α''}}
|-
!style="text-align:left"| {{math|0}}
| {{math|0}}
| {{math|0}}
|| {{math|0}}
|| {{math|0}}
|-
!style="text-align:left"| {{math|1}}
| {{math|0}}
| {{math|1}}
|| {{math|''α''}}
|| {{math|1 + ''α''}}
|-
!style="text-align:left"| {{math|''α''}}
|| {{math|0}}
|| {{math|''α''}}
|| {{math|1 + ''α''}}
|| {{math|1}}
|-
!style="text-align:left"| {{math|1 + ''α''}}
|| {{math|0}}
|| {{math|1 + ''α''}}
|| {{math|1}}
|| {{math|''α''}}
|}
|
{| class="wikitable"
|+ Division <math>x \div y</math>
|-
! style="width:24%;" {{diagonal split header|{{math|''x''}}|{{math|''y''}}}} !! style="width:20%;"| {{math|1}} !! style="width:20%;"| {{math|''α''}} !! style="width:20%;"| {{math|1 + ''α''}}
|-
!style="text-align:left"| {{math|0}}
| {{math|0}}
|| {{math|0}}
|| {{math|0}}
|-
!style="text-align:left"| {{math|1}}
| {{math|1}}
|| {{math|1 + ''α''}}
|| {{math|''α''}}
|-
!style="text-align:left"| {{math|''α''}}
|| {{math|''α''}}
|| {{math|1}}
|| {{math|1 + ''α''}}
|-
!style="text-align:left"| {{math|1 + ''α''}}
|| {{math|1 + ''α''}}
|| {{math|''α''}}
|| {{math|1}}
|}
|}
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 <math>x</math> by <math>y</math>, the values of <math>x</math> must be read in the left column, and the values of <math>y</math> in the top row. (Because <math>0 \cdot z = 0</math> for every <math>z</math> in every [[Ring (mathematics)|ring]] the [[division by 0]] has to remain undefined.) From the tables, it can be seen that the additive structure of <math>\mathrm{GF}(4)</math> is isomorphic to the [[Klein four-group]], while the non-zero multiplicative structure is isomorphic to the group <math>Z_3</math>.

The map
<math display="block"> \varphi:x \mapsto x^2</math>
is the non-trivial field automorphism, called the [[#Frobenius automorphism and Galois theory|Frobenius automorphism]], which sends <math>\alpha</math> into the second root <math>1+\alpha</math> of the above-mentioned irreducible polynomial <math>X^2+X+1</math>.