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, contains classes: Galois thought and Module conception. subject matters coated within the first direction are classical formulation for recommendations of cubic and quartic equations, classical quantity idea, commutative algebra, teams, and Galois conception. subject matters 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 kinds, and Grobner bases.
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Extra resources for Advanced Modern Algebra, Part 1
A polynomial has only finitely many nonzero coefficients. The zero polynomial, denoted by O' = 0, is the sequence 17 = (0, 0, 0, ... ). Definition. If O' = (so, s 1, ... , sn, 0, 0, ... ) is a nonzero polynomial, then there is n ;:::: 0 with Sn ;/; 0 and Si = 0 for all i > n. We call Sn the leading coefficient of 17, we call n the degree of 17, and we denote the degree by n If the leading coefficient Sn = deg(<7). = 1, then 17 is called monic. The zero polynomial 0 does not have a degree because it has no nonzero coefficients.
8. A nonzero commutative ring R is a domain if and only if the product of any two nonzero elements of R is nonzero. Proof. ab = ac if and only if a(b - c) = 0. • It follows easily that a Boolean ring 2X is not a domain if X has at least two elements. Elements a, b E R are called zero divisors if ab domains have no zero divisors. = 0 and a # 0, b # 0. 9. The commutative ring Zm is a domain if and only if m is prime. Proof. If m is not prime, then m = ab, where 1 < a, b < m; hence, both [aJ and [bJ are not zero in Zm, yet [a][bJ = [mJ = [OJ.
7 + 5). 7 + 0). 7 + 6 = 251, 251 . 7 + 6 = 1763, (((0. 7 + 5). 7 + 0). 7 + 6). 7 + 6 = 1763, 1763 . 7 + 4 = 12345, ((((0. 7 + 5). 7 + 0). 7 + 6). 7 + 6). 7 + 4 = 12345. Expanding and collecting terms gives 5 . 74 + 0 . 73 + 6 . 72 + 6 . 7 + 4 = 12005 + 0 + 294 + 42 + 4 = 12345. We have written 12345 in "base 7:" it is 50664. ~ This idea works for any integer b 2 2. 30. If b 2 2 is an integer, then every positive integer h has an expression in base b: there are unique integers di with 0 :::; di < b such that h = dkbk + dk-1bk-I +···+do.