Advanced Modern Algebra, Part 1, 3rd Edition by Joseph J. Rotman

By Joseph J. Rotman

This re-creation, now in components, has been considerably reorganized and plenty of sections were rewritten. this primary half, designed for a primary yr of graduate algebra, involves classes: Galois idea and Module idea. subject matters lined within the first path are classical formulation for ideas of cubic and quartic equations, classical quantity conception, commutative algebra, teams, and Galois conception. issues within the moment path are Zorn's lemma, canonical types, 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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Define a relation= on Rx Rx, where Rx is the set of all nonzero elements in R, by (a, b) = (c, d) if ad = be. We claim that = is an equivalence relation. Verifications of reflexivity and symmetry are straightforward; here is the proof of transitivity. If (a, b) = (c, d) and (c, d) = (e, f), then ad = be and cf = de. But ad= be gives adj= b(cf) = bde. Canceling d, which is nonzero, gives af =be; that is, (a,b) = (e,f). Denote the equivalence class of (a, b) by [a, bl, define Fas the set of all equivalence classes, and equip F with the following addition and multiplication (if we pretend that [a, bl is the fraction a/b, then these are just the familiar formulas): [a, bl + [c, dl = [ad+ be, bd] [a, b][c, dl = [ac, bdl and (since b =fa 0 and d =fa 0, we have bd =fa 0 because Risa domain, and so the formulas make sense).

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. 10 Notation. If Risa commutative ring, then R[[x]] denotes the set of all formal power series over R, and R[x] ~ R[[x]] denotes the set of all polynomials over R.

In each case, the key idea is to solve a problem by replacing numbers by their remainders. 24. (i) If a is in Z, then a 2 = 0, 1, or 4 mod 8. 23(iii), a2 = r 2 mod 8, and so it suffices to look at the squares of the remainders. 1. Squares mod 8. 1 that only 0, 1, or 4 can be a remainder after dividing a perfect square by 8. (ii) n = 1003456789 is not a perfect square. Since 1000 = 8 · 125, we have 1000 = 0 mod 8, and so n = 1003456789 = 1003456 · 1000 + 789 =789 mod 8. Dividing 789 by 8 leaves remainder 5; that is, n perfect square, then n = 0, 1, or 4 mod 8.

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