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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