The Square Root and the Relations of Order

Oswald Veblen
1906 Transactions of the American Mathematical Society  
One of the most obvious discriminations between positive and negative numbers is that the former possess square roots in the field of reals while the latter do not. This distinction, however, has not yet been used in any of the current systems of postulates. On the contrary, an order relation, <, is usually introduced as an undefined symbol. Though the existence of a square root cannot be deduced from order relations without the use of a continuity assumption of some sort, it turns out to be
more » ... y easy to infer the order relations from postulates about the existence of a square root. Suppose we are given a field, defined by one of the numerous sets of postulates published in the Transactions by Huntington, Moore and Dickson. This field contains a unique mark, 0, such that a;-f-0 = a; = 0 + a: for every mark x, and a unique mark, 1, such that lx = x for every mark x. A mark, a, is called a square if there exists a mark, x, such that xx = a. If not a square, a mark is called a not-square. Now add the postulates : f a) The mark -1 is a not-square. ß) If marks x and y are not-squares, then x + y is a not-square. The postulate a) shows that the marks 2 and 0 are distinct and thus that division by 2 is possible.
doi:10.2307/1986433 fatcat:4qfpyegiqvek3gfqaqz2it4as4