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**Additional resources for A Theory of Differentiation in Locally Convex Spaces / Memoirs No. 212**

**Example text**

The above proof can be analysed as follows, (i) The ring in question has divisors of zero—that is, 0 can be expressed as a product of non-zero elements in the ring—if and only if m is composite, (ii) There can be no divisors of zero in any ring which is afieldor a part of a field, (iii) A commutative ring that has a unit element and no divisors of zero—that is, an integral domain—is necessarily a field if it is finite. ) In the ring of integers modulo 7, 3 is the reciprocal of 5, since 3 x 5 = 1 (mod 7).

172. , ( M - l ) 2 give exactly (M+l)/2 different remainders on division by M. If M is odd but not a prime, state, with reasons, whether or not the number of different remainders is necessarily (M+l)/2, and, if not, whether it may be larger. Find the integers between 1 and 12 which are not congruent to squares modulo 13. 173. Show that in the field of rational numbers the equations 4x + 3y+ z = 1, 2x+ y + Az = 1, x - 5 s = 1, EXERCISES 174-177 45 have only one solution, but that in the field of integers modulo 7 they have more than one solution.

196. -\-b2 and b2 + a2 counting as the same). Express 14,645 as the sum of squares of two natural numbers in four ways. 197. If a, b, c are any three vectors, prove that aA(bAc) = ( a . c ) b - ( a . b ) c . Show that the condition a A c = 0 is sufficient for aA(bAc) = (aAb)AC. Determine whether or not it is a necessary condition. 198. Let rational numbers be written in the formp/q, where/) and q are integers with no common factor. Prove that in any interval there are rational numbers with q even and also rational numbers with q odd.