By Joseph J. Rotman
This new version, now in elements, has been considerably reorganized and lots of sections were rewritten. this primary half, designed for a primary yr of graduate algebra, includes classes: Galois conception and Module thought. subject matters coated within the first path are classical formulation for recommendations of cubic and quartic equations, classical quantity conception, commutative algebra, teams, and Galois conception. themes within the moment path are Zorn's lemma, canonical varieties, internal product areas, different types and bounds, tensor items, projective, injective, and flat modules, multilinear algebra, affine types, and Grobner bases.
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Extra info for Advanced Modern Algebra, Part 1
N is not divisible by the square of any prime), then fo is irrational. (ii) Prove that an integer m 2: 2 is a perfect square if and only if each of its prime factors occurs an even number of times. 7. '12 is irrational. Hint. '12 can be written as a fraction in lowest terms. 8. ) = gcd(ab, ac). (We must assume that a > 0 lest Hint. Show that if k is a common divisor of ab and ac, then k I agcd(b, c). 9. (i) Show that if d is the greatest common divisor of ai, a2, ... , an, then d where ti is in Z for 1 :::; i :::; n.
Find the units in Zn and compute their multiplicative inverses. 14. Prove that Q has no proper subfields. 15. Prove that every domain R with a finite number of elements must be a field. ) Hint. If Rx denotes the set of nonzero elements of Rand r E Rx, apply the Pigeonhole Principle (If X is a finite set, then the following are equivalent for f: X -t X: f is an injection; f is a bijection; f is a surjection) after proving that multiplication by r is an injection Rx -t Rx. 16. It may seem more natural to define addition in the Boolean ring 2X as union rather than symmetric difference.
For example, the number 2 is a unit in Q, for ~ lies in Q and 2 x ~ = 1, but 2 is not a unit in Z, because there is no integer v with 2v = 1. In fact, the only units in Z are 1 and -1. What are the units in Zm? 11. If a is an integer, then [a] is a unit in Zm if and only if a and m are relatively prime. In fact,. if sa + tm = 1, then [aJ- 1 = (s]. Proof. 34. 12. If p is prime, then every nonzero [a] in Zp is a unit. Proof. If 1 ~a< p, then gcd(a,p) = 1. • 7 It is easy to prove the Leibniz formula by induction on n, but it is not a special case of the Binomial Theorem.