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 ≤ .

### First course in theory of numbers by H.N. Wright

by William

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