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- W2004750140 abstract "Given the operations of subtraction and reciprocaltaking we show that the operation of multiplication is determined in any field. The amusing difference is that for characteristic 2 it is not given by a formula while for other characteristics, it is. The identity xy = ((x + y 2)-1 (x + y + 2)1)-' ((x-y-2)-1(x-y+ 2)y )' shows how multiplication of real numbers is determined from the operation of subtraction and the taking of reciprocals. In more colorful language: a machine which can perform subtractions and take reciprocals can be programmed to multiply. (The cases of 0 denominators being easily interpreted.) The formula, (1), is in fact valid in a much wider setting. It holds (with the previously mentioned conventions) in any field of characteristic not 2. In a field of characteristic 2, however, it is totally useless and meaningless and the question arises as to what happens in such fields. The answer is somewhat surprising. In a field of characteristic 2: (2) The operations of subtraction and taking reciprocals do determine multiplication, but there is no formula for the product in terms of iterated subtractions and reciprocals! To prove (2) observe first the identity 2 = x + (4) 1 1 x 1+ x which shows that the square of every element is determined (the square of 0 being 0 and of 1 being 1). Received by the editors February 3, 1970. AMS 1969 subject classifications. Primary 1310; Secondary 0504" @default.
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- W2004750140 date "1971-02-01" @default.
- W2004750140 modified "2023-09-27" @default.
- W2004750140 title "Multiplication from other operations" @default.
- W2004750140 doi "https://doi.org/10.1090/s0002-9939-1971-0269629-x" @default.
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