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Elements of Number Theory

Tác giả: John Stillwell
Loại tài liệu: Sách điện tử
Nội dung tóm tắt: Xem chi tiết
This book is intended to complement my Elements oi Algebra, and it is similarly motivated by the problem of solving polynomial equations. However, it is independent of the algebra book, and probably easier. In Elements oi Algebra we sought solution by radicals, and this led to the concepts of fields and groups and their fusion in the celebrated theory of Galois. In the present book we seek integer solutions, and this leads to the concepts of rings and ideals which merge in the equally celebrated theory of ideals due to Kummer and Dedekind. Solving equations in integers is the central problem of number theory, so this book is truly a number theory book, with most of the results found in standard number theory courses. However, numbers are best understood through their algebraic structure, and the necessary algebraic concepts­ rings and ideals-have no better motivation than number theory. The first nontrivial examples of rings appear in the number theory of Euler and Gauss. The concept of ideal-today as routine in ring the­ ory as the concept of normal subgroup is in group theory-also emerged from number theory, and in quite heroic fashion. Faced with failure of unique prime factorization in the arithmetic of certain generalized inte­ gers , Kummer created in the 1840s a new kind of number to overcome the difficulty. He called them ideal numbers because he did not know exactly what they were, though he knew how they behaved.

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Thông tin chi tiết

Dạng tài liệu: Bản điện tử
Tác giả: John Stillwell
Nhà xuất bản: Springer
Năm xuất bản: 2003
Mô tả vật lý:
Từ khóa: Euclidean algorithm, Prime number, number theory

Từ khóa