By M Droste, R. Gobel

Comprises 25 surveys in algebra and version thought, all written via prime specialists within the box. The surveys are established round talks given at meetings held in Essen, 1994, and Dresden, 1995. every one contribution is written in this kind of manner as to spotlight the information that have been mentioned on the meetings, and in addition to stimulate open learn difficulties in a kind available to the entire mathematical group.

The themes comprise box and ring thought in addition to teams, ordered algebraic constitution and their courting to version conception. a number of papers take care of endless permutation teams, abelian teams, modules and their family and representations. version theoretic points comprise quantifier removing in skew fields, Hilbert's seventeenth challenge, (aleph-0)-categorical buildings and Boolean algebras. in addition symmetry questions and automorphism teams of orders are coated.

This paintings includes 25 surveys in algebra and version idea, every one is written in the sort of means as to focus on the tips that have been mentioned at meetings, and likewise to stimulate open study difficulties in a sort obtainable to the complete mathematical group.

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**Example text**

Many problems on ordinary primes ean be reformulated for Gaussian primes. Gaussian integers, a + bi where a, bare integers and i 2 = -1, behave like ordinary integers in the sense that there is unique factorization (apart from order, units (± 1, ± i) and associates; the assoeiates of 7, for example, are 7, - 7, 7i, and - 7i). Primes of the form 4k - 1 are still primes (3, 7,11 , 19,23, ... ) but the other ordinary primes ean be faetored into Gaussian pnmes : 13 = + i)(1 2 = (2 + 3i)(2 (1 - i), - 3i), + i)(2 - i) = -(2i - 1)(2i + 1), ete.

J. Lee, On divisibility by ni ne of the sums of even amicable pairs, Math. Comput. 23 (1969) 545-548; MR 40 #1328. E. J. Lee and J. S. Madachy, The history and discovery of amicable numbers, part 1, J. Recreationa/ Math. 5 (1972) 77 -93; part 2, ibid. 153-173; part 3, ibid. 231-249. O. are, Number Theory and its History, McGraw-Hill, New York, 1948, p. 89. Carl Pomerance, On the distribution of amicable numbers, J. reine angeH'. Math. 293/294 (1977) 217-222; II ibid. (1981). P. Poulet, 43 new couples of amicable numbers, Scripta Math.

E •• ce I! /... • .......... , , •• , ~ .. \" ......... ". ,I ...... ,... e#I .. 'iM · . ~ ~ ~ Figure 5. The Eisenstein Primes. . ,~, 22 Unsolved Problems in Number Theory J. H. Jordan and J. R. Rabung, A conjecture of Paul Erdös concerning Gaussian primes, Math. Comput. 24 (1970) 221-223. Al7. Perhaps the philosopher's stone of number theory is a formula for Pn, or for n(x), or for a necessary and sufficient condition for primality. Wilson's theorem seems to be unique, but even that is useless for computation.