Lester Hill’s “The checking of the accuracy …”, part 10

Construction of finite fields for use in checking

Let $F_\Gamma$ denote a finite algebraic field with $\Gamma$ elements. It is well-known that, for a given $\Gamma$ , all fields $F_\Gamma$ are “simply isomorphic”, and therewith, for our purposes, identical. We shall consequently refer, without restraint, to ”the field $F_\Gamma$ .”

If $p$ is a prime positive integer greater than $1$ , $F_p$ is called, according to the terminology of Section 8, Example 2, a ”primary” field. Explicit addition tables, as was noted in section 8, are hardly required in deal ing with primary fields. The most useful of these fields, in telegraphic checking, are probably $F_{23}$ , $F_{29}$ , $F_{31}$ , and $F_{101}$ . The field $F_{101}$ will be considered in detail in what follows.

The number of elements in a non-primary finite algebraic field
is a power of a prime. If we have

$q=p^k$
where $p$ and $k$ are positive integers greater than $1$ , and $p$ is prime, the non-primary field $F_q$ may be constructed very easily by algebraic extension of the field $F_p$ . Explicit addition table is needed when working with a non-primary field. Otherwise, checking operations are exactly the same as in primary fields.

Example: The field $F_3$ with the elements (marks) $0,1,2$ , has the tables

$\begin{array}{r|*{3}{r}} \multicolumn{1}{c|} +&0&1&2\\\hline {}0&0&1&2\\ {}1&1&2&0\\ {}2&2&0&1\\ \end{array}$
$\begin{array}{r|*{3}{r}} \multicolumn{1}{c|} \cdot &0&1&2\\\hline {}0&0&0&0\\ {}1&0&1&2\\ {}2&0&2&1\\ \end{array}$

By adjoining a root of the equation $x^2=2$ , an equation which is irreducible in $F_3$ , we easily obtain the field $F_9$ with marks

$\alpha j+\beta$
where $\alpha$ and $\beta$ denote elements of $F_3$ . The marks of $F_9$ are thus

$0,1,2,j,j+1,j+2,2j,2j+1,2j+2.$
These marks (elements) are combined, in the rational field operations of $F_9$ , according to the reduction formula $j^2=2$ . If we label the marks of $F_9$ as follows

$\begin{array}{ccccccccc} 0 & 1 & 2 & j & j+1& j+2& 2j& 2j+1& 2j+2\\ 0 & 1 & a & b & c & d & e & f & g\\ \end{array}$
the addition and multiplication tables of the field are given as in
Section 8, Example 1.

In a like manner, $F_{27}$ can be obtained from $F_3$ by adjunction of a root of the equation $x^3=x+1$ , which is irreducible in $F_3$ and $F_9$ . The marks (elements) of $F_{27}$ are

$\alpha j^2+\beta j+\gamma,$
where $\alpha,\beta,\gamma$ are elements of $F_3$ . They are combined, in the rational operations of $F_{27}$ according to the reduction formula $j^3=j+1$ .

Example: The field $F_2$ with the elements (marks) $0,1$ , has the tables

$\begin{array}{r|*{2}{r}} \multicolumn{1}{c|} + &0&1\\ \hline {}0&0&1\\ {}1&1&0\\ \end{array}$
$\begin{array}{r|*{2}{r}} \multicolumn{1}{c|} \cdot &0&1\\ \hline {}0&0&0\\ {}1&0&1\\ \end{array}$

By adjunction of a root of the equation $x^5=x^2+1$ , which is irreducible in the fields $F_2$ , $F_4$ , $F_8$ and $F_{16}$ , we easily obtain the field $F_{32}$ . The marks of $F_{32}$ are

$\alpha j^4+\beta j^3+\gamma j^2+\delta j+\epsilon,$
where $\alpha,\beta,\gamma, \delta,\epsilon$ are elements of $F_2$ ; and these $32$ marks are combined, in the rational operations of $F_{32}$ , according to the reduction formula $j^5=j^2+1$ .

Example: The field $F_5$ with the elements (marks) $0,1,2,3,4$ , has the tables

$\begin{array}{r|*{5}{r}} \multicolumn{1}{c|} +&0&1&2&3&4\\\hline {}0&0&1&2&3&4\\ {}1&1&2&3&4&0\\ {}2&2&3&4&0&1\\ {}3&3&4&0&1&2\\ {}4&4&0&1&2&3\\ \end{array}$
$\begin{array}{r|*{5}{r}} \multicolumn{1}{c|} \cdot &0&1&2&3&4\\\hline {}0&0&0&0&0&0\\ {}1&0&1&2&3&4\\ {}2&0&2&4&1&3\\ {}3&0&3&1&4&2\\ {}4&0&4&3&2&1\\ \end{array}$

By adjoining a root of the equation $x^2=2$ , which is irreducible in $F_{5}$ , we readily obtain the field $F_{25}$ . The marks of $F_{25}$ are

$\alpha j+\beta ,$
where $\alpha,\beta$ are elements of $F_5$ ; and these $25$ marks are combined, in the rational operations of $F_{25}$ , according to the reduction formula $j^2=2$ .

Of the non-primary fields, $F_{25}$ , $F_{27}$ , $F_{32}$ are probably those which are most amenable to practical application in telegraphic checking.

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Yet Another Mathblog

Lester Hill’s “The checking of the accuracy …”, part 10

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