**Construction of finite fields for use in checking**

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

If is a prime positive integer greater than , 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 , , , and . The field 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

where and are positive integers greater than , and is prime, the non-primary field may be constructed very easily by algebraic extension of the field . 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 with the elements (marks) , has the tables

By adjoining a root of the equation , an equation which is irreducible in , we easily obtain the field with marks

where and denote elements of . The marks of are thus

These marks (elements) are combined, in the rational field operations of , according to the reduction formula . If we label the marks of as follows

the addition and multiplication tables of the field are given as in

Section 8, Example 1.

In a like manner, can be obtained from by adjunction of a root of the equation , which is irreducible in and . The marks (elements) of are

where are elements of . They are combined, in the rational operations of according to the reduction formula .

Example: The field with the elements (marks) , has the tables

By adjunction of a root of the equation , which is irreducible in the fields , , and , we easily obtain the field . The marks of are

where are elements of ; and these marks are combined, in the rational operations of , according to the reduction formula .

Example: The field with the elements (marks) , has the tables

By adjoining a root of the equation , which is irreducible in , we readily obtain the field . The marks of are

where are elements of ; and these marks are combined, in the rational operations of , according to the reduction formula .

Of the non-primary fields, , , are probably those which are most amenable to practical application in telegraphic checking.