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A First Course in Abstract Algebra: Rings, Groups, and by Marlow Anderson PDF

By Marlow Anderson

ISBN-10: 1482245523

ISBN-13: 9781482245523

ISBN-10: 1482245531

ISBN-13: 9781482245530

A part of the PWS complicated arithmetic sequence, this article contains chapters on polynomials and factoring, targeted factorization, ring homomorphisms and beliefs, and constructibility difficulties and box extensions.

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Extra resources for A First Course in Abstract Algebra: Rings, Groups, and Fields

Example text

Consider the following multiplication tables for our examples Z5 and Z6 . · [0] [1] [2] [3] [4] [0] [0] [0] [0] [0] [0] · [0] [1] [2] [3] [4] [5] multiplication tables Z5 and Z6 [1] [0] [1] [2] [3] [4] [2] [0] [2] [4] [1] [3] [3] [0] [3] [1] [4] [2] [4] [0] [4] [3] [2] [1] [0] [0] [0] [0] [0] [0] [0] [1] [0] [1] [2] [3] [4] [5] [2] [0] [2] [4] [0] [2] [4] [3] [0] [3] [0] [3] [0] [3] [4] [0] [4] [2] [0] [4] [2] [5] [0] [5] [4] [3] [2] [1] Notice the remarkable fact that in Z5 , every element (other than [0]) has a multiplicative inverse.

Then express this gcd as a linear combination of 34 and 21. (b) Now do the same for 2424 and 772. (c) Do the same for 2007 and 203. (d) Do the same for 3604 and 4770. 2. (a) Prove that gcd(a, b) divides a − b. This sometimes provides a short cut in finding gcds. (b) Use this to find gcd(1962, 1965). (c) Now find gcd(1961, 1965). (d) Find the gcds in Exercise 1 using this short cut. 3. Prove that the set of all linear combinations of a and b are precisely the multiples of gcd(a, b). 4. Two numbers are said to be relatively prime if their gcd is 1.

1, 3, 7, 11, . }. The next theorem provides a very useful way of determining when two integers are in the same residue class. Indeed, we will use this characterization more often than the definition itself. 2 Two integers, x and y, have the same residue (mod m) if and only if x−y = km for some integer k. Proof: First, suppose x ≡ y (mod m). Then x = k1 m + r, and y = k2 m + r for some integers k1 and k2 and 0 ≤ r < m. But then x − y = (k1 − k2 )m. Conversely, suppose x−y = km, for some integer k with x = k1 m+r1 and y = k2 m+r2 , where 0 ≤ r1 < m and 0 ≤ r2 < m.

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A First Course in Abstract Algebra: Rings, Groups, and Fields by Marlow Anderson


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