# New PDF release: First course in theory of numbers

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By H.N. Wright

ISBN-10: 0486606740

ISBN-13: 9780486606743

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Extra resources for First course in theory of numbers

Example text

S. Barnes (1874–1953), H. Behrbohm, E. Berg, A. T. Brauer (1894–1985), H. Chatland, H. Davenport (1907–1969), L. E. Dickson (1874–1954), P. Erd¨os (1913–1996), H. A. Heilbronn (1908–1975), N. 2 Examples of Euclidean Domains 35 L. K. Hua, K. Inkeri, J. F. Keston, C. Ko, S. H. Min, A. Oppenheim, O. Perron (1880–1975), L. R´edei, R. Remak (1888–1942), L. Schuster, W. T. Sheh, and H. P. F. Swinnerton-Dyer. The final step was taken in 1950 by Chatland and Davenport [4], who established the following two theorems.

3). √ √ Express 2 + 8 −5 as a product of irreducibles in Z + Z −5. In how many ways can this be done? √ √ Prove that −6 is not a prime in Z + Z −6. √ √ Prove that −6 is an irreducible in Z + Z −6. √ Prove that Z + Z −6 is not a principal ideal domain. √ −6 that is not principal. Give an example of an ideal in Z + Z √ √ Prove that √10 is not a prime in Z + Z 10. √ Prove that 10 is√an irreducible in Z + Z 10. Prove that Z + Z 10 is not a principal √ideal domain. Give an example of an ideal in Z + Z 10 that is not principal.

8 The integral domain Z + Z m is Euclidean with respect to φm for m = 2, 3, 6. Proof: m = 2, 3. Let x, y ∈ Q. We choose a, b ∈ Z such that 1 1 |x − a| ≤ , |y − b| ≤ . 2 2 As (x − a)2 ≥ 0 and m(y − b)2 ≥ 0, we have 3 |(x − a)2 − m(y − b)2 | ≤ max(|x − a|2 , m|y − b|2 ) ≤ . 2. √ m = 6. Suppose that Z + Z 6 is not Euclidean with respect to φ6 . 2, there exist r, s ∈ Q such that √ √ φ6 ((r + s 6) − (x + y 6)) ≥ 1 for all x, y ∈ Z; that is, |(r − x)2 − 6(s − y)2 | ≥ 1 for all x, y ∈ Z. CB609-02 CB609/Alaca & Williams August 27, 2003 16:51 36 Euclidean Domains We can choose 1 = ±1 and u 1 ∈ Z such that 0≤ and Char Count= 0 2 1 2 1r + u1 ≤ 2s 1 + u2 ≤ .