Gauss Lemma and Law of Quadratic Reciprocity

The papers [20], [10], [9], [11], [4], [1], [2], [17], [8], [19], [7], [16], [13], [21], [22], [5], [18], [3], [15], [6], and [23] provide the terminology and notation for this paper. For simplicity, we adopt the following convention: i, i1, i2, i3, j, a, b, x denote integers, d, e, n denote natural numbers, f , f ′ denote finite sequences of elements of Z, g, g1, g2 denote finite sequences of elements of R, and p denotes a prime number. We now state two propositions: (1) If i1 | i2 and i1 | i3, then i1 | i2 − i3. (2) If i | a and i | a− b, then i | b. Let us consider f . The functor PZ(f) yields a function from Z into Z and is defined by the condition (Def. 1).