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Chapter 7 Introducing Algebraic Number Theory (Commutative Algebra 1)

Book Id:WPLBN0000673792 Format Type:PDF eBook File Size:410.82 KB Reproduction Date:2005

Chapter 7 Introducing Algebraic Number Theory (Commutative Algebra 1). (n.d.). Chapter 7 Introducing Algebraic Number Theory (Commutative Algebra 1). Retrieved from http://gutenberg.us/

Description
Mathematics document containing theorems and formulas.

Excerpt
Excerpt: The general theory of commutative rings is known as commutative algebra. The main applications of this discipline are to algebraic number theory, to be discussed in this chapter, and algebraic geometry, to be introduced in Chapter 8. Techniques of abstract algebra have been applied to problems in number theory for a long time, notably in the effort to prove Fermat?s Last Theorem. As an introductory example, we will sketch a problem for which an algebraic approach works very well. If p is an odd prime and p = 1 mod 4, we will prove that p is the sum of two squares, that is, p can be expressed as x2 + y2 where x and y are integers. Since p?1 2 is even, it follows that -1 is a quadratic residue (that is, a square) mod p. [Pair each of the numbers 2,3,. . . ,p ? 2 with its inverse mod p and pair 1 with p ? 1 = ?1 mod p. The product of the numbers 1 through p ? 1 is, mod p...