1.1. Group s and fields 3 c. C , the field of complex numbers . d. Z 2 denote s th e field consistin g o f th e tw o element s 0 , 1 , with ad - dition an d multiplication define d mo d 2 (so that 1 + 1 = 0 ) . More generally , i f p i s a prime, th e se t Z p o f residue classe s mo d /?, with addition and multiplication mo d /?, is a field. (Se e exercis e exl.1.4.) The fields Q , R , an d C ar e familiar , an d ar e th e mos t commonl y used. Les s familiar, ye t very useful, ar e the finite fields Z p , mentione d above. Also important ar e extensions o f a given field, se e A.5.5. EXERCISES FOR SECTION 1. 1 exl.1.1 Verif y that Sn i s not commutative if n 2. exl.1.2 Sho w that if F is a field, then 0 • a = 0 for all a e F , and if ab = 0 , then a = 0 or b = 0 . exl.1.3 Verif y that Z3 = {0,1,2 } is a field if addition and multiplication are defined mo d 3 , i.e., w e add and multiply a s usual, an d if the result is 3 , subtract 3 thus 1 + 1 = 2 but 2 x 2 = 1 . Why i s Z 4 , define d similarly—a s th e se t {0,1,2,3 } wit h additio n an d multiplication defined mod 4—not a field? exl.1.4 Le t p 1 be a positive integer. Recal l tha t two integers m , n ar e congruent mod p, writte n n = m (mo d p), i f n — m is divisible by p. Thi s is an equivalence relation (se e appendix A.l). Fo r m G Z, denote by m the coset (equivalence class) of m, that is, the set of all integers n such that n = m (mod p). a. Ever y integer is congruent mod p to one of the numbers [0,1,...,/? — 1]. In other words, there is a 1-1 correspondence between Zp, th e set of cosets (mod p), an d the integers [0,1,...,/? — 1]. b. A s i n subsectio n 1.2. 4 above , w e defin e th e quotient ring Z p = Z/pZ (both notation s ar e common) a s the spac e whos e element s ar e the coset s (mod p) i n Z, an d define additio n an d multiplication by : m + n — ( m + n) and m-n — m^fi. Prov e tha t th e additio n an d multiplicatio n s o defined ar e associative, commutative, and satisfy the distributive law. c. Prov e that Zp, endowed with these operations, is a field if and only if p is prime.

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