The square root and the relations of order
Oswald Veblen · Transactions of the American Mathematical Society · 1906
One of the most obvious discritninations 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 beeln 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 very easy to infer the order relations from postulates about the existence of a square root. Suppose we are given a field, defined by olne of the numerous sets of postulates published in the T r a n s a c t i o n s by HUNTINGTON, MOORE and DICKSON.. This field contailns a unique mark, 0, such that x + 0 = x = 0 + x 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: t a) The mark 1 is a not-square. 13) 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. Theorenm 1. If x and x' are such that x + x' = 0, then either x or x' is a square, whereas the other is a not-square. Proof. If x and x' were both not-squares, then by 18), x + x' would be a not-square, whereas zero i's a square. The mark x' = 1 x and hence by the theorem that the product of a square and a not-square is a not-square, if one of the marks x, x' is a square the other is a not square. Theor em 2. If x and y are squares, then x + y is a square.