By Frederick M. Goodman
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Additional resources for Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition)
For (b), a b is divisible by n if, and only if, b a is divisible by n. b c/ is divisible by n. mod n/g D fa C k n W k 2 Zg: Note that this is just the set of all integer points that land at the same place as a when the number line is wrapped around the clock face. The set Œa is called the residue class or congruence class of a modulo n. 7. a/ the unique number r such that 0 Ä r < n and a r is divisible by n. 7). a/ is the unique element of Œa that lies in the interval f0; 1; : : : ; n 1g. a/ is the label on the circumference of the clock face at the point where a falls when the number line is wrapped around the clock face.
Therefore, since KŒx has infinitely many irreducible polynomials, it has irreducible polynomials of arbitrarily large degree. 11. Let’s recall the process of long division of polynomials by working out an example. Let p D 7x 5 C 3x 2 C x C 2 and d D 5x 3 C 2x 2 C 3x C 4. d /. The first contribution to q is 75 x 2 , and p 7 2 x dD 5 The next contribution to q is p 7 . x2 5 7 . x2 5 14 25 x, 14 x/ d D 25 The next contribution to q is p 14 4 x 5 14 x 25 77 125 , 21 3 x 5 13 2 x C x C 2: 5 and 77 3 x 25 23 2 81 x C x C 2: 25 25 and 77 39 2 636 558 /d D x C xC : 125 125 125 125 Thus, qD 7 2 x 5 14 x 25 77 125 50 1.
The operations of addition and multiplication of polynomials can be recast as operations on such functions. It is straightforward (but not especially enlightening) to carry this out. 1. 5x 7 C 9x 3 C x 2 C 2x C 8/ D 10 x 10 C 20 x 8 C 25 x 7 C 18 x 6 C 2 x 5 C 40 x 4 C 65 x 3 C 13 x 2 C 42 x C 40: K can be regarded as subset of KŒx, and the addition and multiplication operations on KŒx extend those on K; that is, for any two elements in K, their sum and product as elements of K agree with their sum and product as elements of KŒx.
Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition) by Frederick M. Goodman