EAN: 9781441930668

UNDERGRADUATE TEXTS IN MATHEMATICS ELEMENTS O...

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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