Editing Finite field arithmetic (section)

==Addition and subtraction==
Addition and subtraction are performed by adding or subtracting two of these polynomials together, and reducing the result modulo the characteristic.

In a finite field with characteristic 2, addition modulo 2, subtraction modulo 2, and XOR are identical. Thus,

{| class="wikitable"
|-
! Polynomial
| (''x''<sup>6</sup> + ''x''<sup>4</sup> + ''x'' + 1) + (''x''<sup>7</sup> + ''x''<sup>6</sup> + ''x''<sup>3</sup> + ''x'') = ''x''<sup>7</sup> + ''x''<sup>4</sup> + ''x''<sup>3</sup> + 1
|-
! Binary
| {01010011} + {11001010} = {10011001}
|-
! Hexadecimal
| {53} + {CA} = {99}
|}

Under regular addition of polynomials, the sum would contain a term 2''x''<sup>6</sup>. This term becomes 0''x''<sup>6</sup> and is dropped when the answer is reduced modulo 2.

Here is a table with both the normal algebraic sum and the characteristic 2 finite field sum of a few polynomials:

{| class="wikitable"
 |-
 ! rowspan=2 | ''p''<sub>1</sub>
 ! rowspan=2 | ''p''<sub>2</sub>
 ! colspan=2 | ''p''<sub>1</sub> + ''p''<sub>2</sub> under...
 |-
 ! [[Polynomial ring|K[''x'']]]
 ! GF(2<sup>''n''</sup>)
 |- 
 | ''x''<sup>3</sup> + ''x'' + 1
 | ''x''<sup>3</sup> + ''x''<sup>2</sup>
 | 2''x''<sup>3</sup> + ''x''<sup>2</sup> + ''x'' + 1
 | ''x''<sup>2</sup> + ''x'' + 1
 |-
 | ''x''<sup>4</sup> + ''x''<sup>2</sup>
 | ''x''<sup>6</sup> + ''x''<sup>2</sup>
 | ''x''<sup>6</sup> + ''x''<sup>4</sup> + 2''x''<sup>2</sup>
 | ''x''<sup>6</sup> + ''x''<sup>4</sup>
 |-
 | ''x'' + 1
 | ''x''<sup>2</sup> + 1
 | ''x''<sup>2</sup> + ''x'' + 2
 | ''x''<sup>2</sup> + ''x''</tr>
 |-
 | ''x''<sup>3</sup> + ''x''
 | ''x''<sup>2</sup> + 1
 | ''x''<sup>3</sup> + ''x''<sup>2</sup> + ''x'' + 1
 | ''x''<sup>3</sup> + ''x''<sup>2</sup> + ''x'' + 1
 |-
 | ''x''<sup>2</sup> + ''x''
 | ''x''<sup>2</sup> + ''x''
 | 2''x''<sup>2</sup> + 2''x''
 | 0
 |}

In computer science applications, the operations are simplified for finite fields of characteristic 2, also called GF(2<sup>''n''</sup>) [[Galois field]]s, making these fields especially popular choices for applications.