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}.