Lester Hill’s “The checking of the accuracy …”, part 11

The field $F_{101}$

All essential points connected with the checking of telegraphic sequences by the methods proposed in this paper may be fully illustrated in one finite field. For our purposes, perhaps the most useful field is $F_{101}$, to which we shall confine our attention in the following sections. The elements of the field $F_{101}$ are the one hundred and one marks\footnote{Hill actually uses the symbol $X$ in place of $100$.} $0$, $1$, $2$, $\dots$, $100$. The operations of addition and multiplication are effected as explained in a previous example; and are abbreviated as suggested. To determine sums and products, we regard the marks of the field momentarily as integers of elementary arithmetic. Thus we have

$\sum_1^n f_i = f_h,\ \ \ \ \ \ (\prod_1^n f_i = f_k),$

the $f_i$ being $n$ marks of $F_{101}$, distinct or not, if, when the $f_i$ are momentarily regarded as integers of elementary arithmetic, the congruence

$\sum_1^n f_i \equiv f_h \pmod{101},\ \ \ \ \ \ (\prod_1^n f_i \equiv f_k \pmod{101}),$

holds. It will not be possible to provide a full multiplication table for the field $F_{101}$. But the following special table will be found convenient.

$\begin{array}{r|rrr} x & x^2 & 1/x & -x\\ 1 & 1 & 1 & 100 \\ 2 & 4 & 51 & 99 \\ 3 & 9 & 34 & 98 \\ 4 & 16 & 76 & 97 \\ 5 & 25 & 81 & 96 \\ 6 & 36 & 17 & 95 \\ 7 & 49 & 29 & 94 \\ 8 & 64 & 38 & 93 \\ 9 & 81 & 45 & 92 \\ 10 & 100 & 91 & 91 \\ 11 & 20 & 46 & 90 \\ 12 & 43 & 59 & 89 \\ 13 & 68 & 70 & 88 \\ 14 & 95 & 65 & 87 \\ 15 & 23 & 27 & 86 \\ 16 & 54 & 19 & 85 \\ 17 & 87 & 6 & 84 \\ 18 & 21 & 73 & 83 \\ 19 & 58 & 16 & 82 \\ 20 & 97 & 96 & 81 \\ 21 & 37 & 77 & 80 \\ 22 & 80 & 23 & 79 \\ 23 & 24 & 22 & 78 \\ 24 & 71 & 80 & 77 \\ 25 & 19 & 97 & 76 \\ 26 & 70 & 35 & 75 \\ 27 & 22 & 15 & 74 \\ 28 & 77 & 83 & 73 \\ 29 & 33 & 7 & 72 \\ 30 & 92 & 64 & 71 \\ 31 & 52 & 88 & 70 \\ 32 & 14 & 60 & 69 \\ 33 & 79 & 49 & 68 \\ \end{array}$

$\begin{array}{r|rrr} x & x^2 & 1/x & -x\\ 34 & 45 & 3 & 67 \\ 35 & 13 & 26 & 66 \\ 36 & 84 & 87 & 65 \\ 37 & 56 & 71 & 64 \\ 38 & 30 & 8 & 63 \\ 39 & 6 & 57 & 62 \\ 40 & 85 & 48 & 61 \\ 41 & 65 & 69 & 60 \\ 42 & 47 & 89 & 59 \\ 43 & 31 & 47 & 58 \\ 44 & 17 & 62 & 57 \\ 45 & 5 & 9 & 56 \\ 46 & 96 & 11 & 55 \\ 47 & 88 & 43 & 54 \\ 48 & 82 & 40 & 53 \\ 49 & 78 & 33 & 52 \\ 50 & 76 & 99 & 51 \\ 51 & 76 & 2 & 50 \\ 52 & 78 & 68 & 49 \\ 53 & 82 & 61 & 48 \\ 54 & 88 & 58 & 47 \\ 55 & 96 & 90 & 46 \\ 56 & 5 & 92 & 45 \\ 57 & 17 & 39 & 44 \\ 58 & 31 & 54 & 43 \\ 59 & 47 & 12 & 42 \\ 60 & 65 & 32 & 41 \\ 61 & 85 & 53 & 40 \\ 62 & 6 & 44 & 39 \\ 63 & 30 & 93 & 38 \\ 64 & 56 & 30 & 37 \\ 65 & 84 & 14 & 36 \\ 66 & 13 & 75 & 35 \\ \end{array}$

$\begin{array}{r|rrr} x & x^2 & 1/x & -x\\ 67 & 45 & 98 & 34 \\ 68 & 79 & 52 & 33 \\ 69 & 14 & 41 & 32 \\ 70 & 52 & 13 & 31 \\ 71 & 92 & 37 & 30 \\ 72 & 33 & 94 & 29 \\ 73 & 77 & 18 & 28 \\ 74 & 22 & 86 & 27 \\ 75 & 70 & 66 & 26 \\ 76 & 19 & 4 & 25 \\ 77 & 71 & 21 & 24 \\ 78 & 24 & 79 & 23 \\ 79 & 80 & 78 & 22 \\ 80 & 37 & 24 & 21 \\ 81 & 97 & 5 & 20 \\ 82 & 58 & 85 & 19 \\ 83 & 21 & 28 & 18 \\ 84 & 87 & 95 & 17 \\ 85 & 54 & 82 & 16 \\ 86 & 23 & 74 & 15 \\ 87 & 95 & 36 & 14 \\ 88 & 68 & 31 & 13 \\ 89 & 43 & 42 & 12 \\ 90 & 20 & 55 & 11 \\ 91 & 100 & 10 & 10 \\ 92 & 81 & 56 & 9 \\ 93 & 64 & 63 & 8 \\ 94 & 49 & 72 & 7 \\ 95 & 36 & 84 & 6 \\ 96 & 25 & 20 & 5 \\ 97 & 16 & 25 & 4 \\ 98 & 9 & 67 & 3 \\ 99 & 4 & 50 & 2 \\ 100 & 1 & 100 & 1 \end{array}$

Squares, reciprocals, negatives

Using the scheme of reciprocals shown in this Table, we may easily perform an rational operations in $F_{101}$.

Example:

Suppose, for example, that we wish to solve the system of equations:

$36x-79y=52,\ \ \ 90x+85y = 98.$

They may be written

$x-79y/36 = 13/9,\ \ \ \ x+17y/18 = 49/45$

and the fractions are quickly evaluated. Thus: $-79/36 = 96$. Determining the fractions in this manner, we write the two equations in the form:

$x+96y=80,\ \ \ \ x+29y=37,$

whence $y=73$ and $x=41$

The modulus $101$ is very convenient to work with. The residue, modulo $101$, of any integer is immediately obvious, at sight of the integer, and is therefore obtained without computation.