# Difference between revisions of "Commutative Presemifields and Semifields"

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=Introduction= | =Introduction= | ||

− | A <span class="definition">semifields</span> | + | A <span class="definition">semifields</span> is a ring with left and right distributivity and with no zero divisor. |

A presemifield with a multiplicative identity is called a <span class="definition">semifield</span>. | A presemifield with a multiplicative identity is called a <span class="definition">semifield</span>. | ||

Any finite presemifield can be represented by <math>\mathbb{S}=(\mathbb{F}_{p^n},+,\star)</math>, | Any finite presemifield can be represented by <math>\mathbb{S}=(\mathbb{F}_{p^n},+,\star)</math>, |

## Revision as of 11:26, 29 August 2019

# Introduction

A semifields is a ring with left and right distributivity and with no zero divisor. A presemifield with a multiplicative identity is called a semifield. Any finite presemifield can be represented by , for a prime, a positive integer, additive group and multiplication linear in each variable.

Two presemifields and are called isotopic if there exist three linear permutation of such that , for any . If then they are called strongly isotopic. Each commutative presemifields of odd order defines a planar DO polynomial and viceversa:

- given let ;
- given let defined by .

Hence two quadratic planar functions are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:

- are CCZ-equivalent if and only if are strongly isotopic;
- for odd, isotopic coincides with strongly isotopic;
- if are isotopic equivalent, then there exists a linear map such that is EA-equivalent to .