This introduction to algebraic number theory via the famous problem of "Fermats Last Theorem" follows its historical development, beginning with the work of Fermat and ending with Kummers theory of "ideal" factorization. The more elementary topics, such as Eulers proof of the impossibilty of x+y=z, are treated in an uncomplicated way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummers theory to quadratic integers and relates this to Gauss''theory of binary quadratic forms, an interesting and important connection that is not explored in any other book.

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Inhaltsangabe1 Fermat.- 1.1 Fermat and his "Last Theorem." Statement of the theorem. History of its discovery.- 1.2 Pythagorean triangles. Pythagorean triples known to the Babylonians 1000 years before Pythagoras.- 1.3 How to find Pythagorean triples. Method based on the fact that the product of two relatively prime numbers can be a square only if both factors are squares.- 1.4 The method of infinite descent.- 1.5 The casen= 4 of the Last Theorem. In this case the proof is a simple application of infinite descent. General theorem reduces to the case of prime exponents.- 1.6 Fermat's one proof. The proof that a Pythagorean triangle cannot have area a square involves elementary but very ingenious arguments.- 1.7 Sums of two squares and related topics. Fermat's discoveries about representations of numbers in the form n = x2+ kn2 for k =1, 2, 3. The different pattern when k = 5.- 1.8 Perfect numbers and Fermat's theorem. Euclid's formula for perfect numbers leads to the study of Mersenne primes 2n ? 1 which in turn leads to Fermat's theorem ap ? a ? 0 mod p. Proof of Fermat's theorem. Fermat numbers. The false conjecture that 232 + 1 is prime.- 1.9 Pell's equation. Fermat's challenge to the English. The cyclic method invented by the ancient Indians for the solution of Ax2+ 1=y2 for given nonsquare A. Misnaming of this equation as "Pell's equation" by Euler. Exercises: Proof that Pell's equation always has an infinity of solutions and that the cyclic method produces them all.- 1.10 Other number-theoretic discoveries of Fermat. Fermat's legacy of challenge problems and the solutions of these problems at the hands of Lagrange, Euler, Gauss, Cauchy, and others.- 2 Euler.- 2.1 Euler and the case n = 3. Euler never published a correct proof that x3+y3?z3 but this theorem can be proved using his techniques.- 2.2 Euler's proof of the casen = 3. Reduction of Fermat's Last Theorem in the case n = 3 to the statement that p2+ 3q2 can be a cube (p and q relatively prime) only if there exist a and b such that p = a3 -9ab2, q = 3a2b - 3b3.- 2.3 Arithmetic of surds. The condition for p2+ 3q2 to be a cube can be written simply as $$p + y\sqrt { - 3} = {\left( {a + b\sqrt { - 3} } \right)^3}$$, that is, $$p + q\sqrt { - 3} $$ is a cube. Euler's fallacious proof, using unique factorization, that this condition is necessary for p2 + 3q2 = cube.- 2.4 Euler on sums of two squares. Euler's proofs of the basic theorems concerning representations of numbers in the forms x2+y2 and x2+ 3y2. Exercises: Numbers of the form x2+ 2y2.- 2.5 Remainder of the proof whenn = 3. Use of Euler's techniques to prove x3+y3?z3.- 2.6 Addendum on sums of two squares. Method for solving p = x2+y2 when p is a prime of the form 4n + 1. Solving p = x2+3y2 and p = x2+ 2y2.- 3 From Euler to Kummer.- 3.1 Introduction. Lagrange, Legendre, and Gauss. 3.2 Sophie Germain's.- theorem. Sophie Germain. Division of Fermat's Last Theorem into two cases, Case I (x,y, and z relatively prime to the exponent p) and Case II (otherwise). Sophie Germain's theorem is a sufficient condition for Case I. It easily proves Case I for all small primes.- 3.3 The casen= 5. Proof that x5+y5?z5. The joint achievement of Dirichlet and Legendre. General technique is like Euler's proof that x3+y3?z3 except that p2 -5q2 a fifth power implies $$p + q\sqrt 5 = {\left( {a + b\sqrt 5 } \right)^5}$$ only under the additional condition 5 q.- 3.4 The casesn = 14 andn = 7. These proofs, by Dirichlet and Lamé respectively, are not explained here. To go further and prove Fermat's Last Theorem for larger exponents clearly requires new techniques. Exercise: Dirichlet's proof of the case n = 14.- 4 Kummer's theory of ideal factors.- 4.1 The events of 1847. Lamé's "proof" of Fermat's Last Theorem. Liouville's objection. Cauchy's attempts at a proof. Kummer's letter to Liouville. Failure of unique factorization. Kummer's new theory of ideal complex numbers.- 4.2 Cyclotomic integers. Basic definitions and operations. Th

Fermat.- Euler.- From Euler to Kummer.- Kummer''s theory of ideal factors.- Fermat''s Last Theorem.- Determination of the class number.- Divisor''s theory for quadratic equations.- Gauss''s theory of binary quadratic forms.- Dirichlet''s class number formula.