# 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.

# mathematics problem 155

A colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

Mathematics Problem, #155

We can represent a triangle with sides of length a, b, c by the ordered triple (a, b, c). Changing the order of the sides doesn’t change the triangle, so (a, b, c), (b, a, c), (b, c, a), (c, b, a), (c, a, b), and (a, c, b) all represent the same triangle. To avoid confusion, let’s agree to write (a, b, c) with a < b < c. We say that a triangle <I (a, b, c) is integral if a, b, and c are integers. How many integral triangles are there with longest side less than or equal to 100 ?

# Mathematics Problem 154

A colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

Mathematics Problem, #154

Find the volume of the intersection of three cylinders, each of radius a, which are centered on the x-axis, the y-axis, and the z-axis. That is, find the volume of the three dimensional region

E = {(x,y,z) | x2 + y2 < a2, y2 + z2 < a2, z2 + x2 < a2}.