William J. LeVeque's Elementary Theory of Numbers PDF

By William J. LeVeque

ISBN-10: 0486150763

ISBN-13: 9780486150765

Publish 12 months note: First released June 1st 1990
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Superb advent for readers with restricted formal mathematical education.

Topics contain Euclidean set of rules and its outcomes, congruences, powers of an integer modulo m, persisted fractions, Gaussian integers, Diophantine equations, extra. conscientiously chosen difficulties incorporated all through, with solutions. basically highschool math wanted. Bibliography.

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However, if K is a field one can normalize each polynomial to make it monic by dividing by LC(f ). Over a ring the leading term need not be invertible. Example 103 Consider f, g ∈ IR[x] as defined in Example 101. We have √ f (x) = 3x7 + 2x4 − 27x3 + 2x + 100, deg(f ) = 7, LC(f ) = 3 and g(x) = 1024x10 + 256x8 + 32x5 + 16x4 + 4x2 + 1, deg(g) = 10, LC(g) = 1024. 35 Lemma 104 Let (K, +, ·) be a field. The polynomials K[x] form a ring with the operations n f (x) + g(x) = max{n,m} m i fi x + i=0 n gi x fi x · i=0 (fi + gi )xi , = i=0 m i f (x) · g(x) = i i=0 n+m gi x i i = i=0 fj gi−j i=0 xi , j=0 where f, g ∈ K[x] and fi = 0 for i > n and gi = 0 for i > m.

Over a ring the leading term need not be invertible. Example 103 Consider f, g ∈ IR[x] as defined in Example 101. We have √ f (x) = 3x7 + 2x4 − 27x3 + 2x + 100, deg(f ) = 7, LC(f ) = 3 and g(x) = 1024x10 + 256x8 + 32x5 + 16x4 + 4x2 + 1, deg(g) = 10, LC(g) = 1024. 35 Lemma 104 Let (K, +, ·) be a field. The polynomials K[x] form a ring with the operations n f (x) + g(x) = max{n,m} m i fi x + i=0 n gi x fi x · i=0 (fi + gi )xi , = i=0 m i f (x) · g(x) = i i=0 n+m gi x i i = i=0 fj gi−j i=0 xi , j=0 where f, g ∈ K[x] and fi = 0 for i > n and gi = 0 for i > m.

Xn } and so the dimension is dimK (Pn ) = n + 1. Alternative bases are easy to give. Since K is a field, xi can be replaced by ai xi for any nonzero ai ∈ K, also linear combinations are possible. So another basis is given by {5, 3x − 1, −x2 , 2x3 + x, . . , xn + xn−1 + xn−2 + · · · + x + 1}, since the degrees are all different and so none can be a linear combination of the others, while using linear algebra we can get every element as a linear combination. Definition 127 (Subspace) Let V be a vector space over the field K.

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Elementary Theory of Numbers by William J. LeVeque


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