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# Chapter7 - Chapter 7 Introducing Algebraic Number...

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Chapter 7 Introducing Algebraic Number Theory (Commutative Algebra 1) 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 x 2 + y 2 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 multiplicative inverse mod p and pair 1 with p 1 ≡ − 1 mod p . The product of the numbers 1 through p 1 is, mod p , 1 × 2 × · · · × p 1 2 × − 1 × − 2 × · · · × − p 1 2 and therefore ( p 1 2 ) ! 2 ≡ − 1 mod p .] If 1 x 2 mod p , then p divides x 2 + 1. Now we enter the ring of Gaussian integers and factor x 2 + 1 as ( x + i )( x i ). Since p can divide neither factor, it follows that p is not prime in Z [ i ], so we can write p = αβ where neither α nor β is a unit. Define the norm of γ = a + bi as N ( γ ) = a 2 + b 2 . Then N ( γ ) = 1 iff γ = ± 1 or ± i iff γ is a unit. (See Section 2.1, Problem 5.) Thus p 2 = N ( p ) = N ( α ) N ( β ) with N ( α ) > 1 and N ( β ) > 1 , so N ( α ) = N ( β ) = p . If α = x + iy , then p = x 2 + y 2 . 1

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2 CHAPTER 7. INTRODUCING ALGEBRAIC NUMBER THEORY Conversely, if p is an odd prime and p = x 2 + y 2 , then p is congruent to 1 mod 4. (If x is even, then x 2 0 mod 4, and if x is odd, then x 2 1 mod 4. We cannot have x and y both even or both odd, since p is odd.) It is natural to conjecture that we can identify those primes that can be represented as x 2 + | d | y 2 , where d is a negative integer, by working in the ring Z [ d ]. But the Gaussian integers ( d = 1) form a Euclidean domain, in particular a unique factorization domain. On the other hand, unique factorization fails for d ≤ − 3 (Section 2.7, Problem 7), so the above argument collapses. [Recall from (2.6.4) that in a UFD, an element p that is not prime must be reducible.] Diﬃculties of this sort led Kummer to invent “ideal numbers”, which later became ideals at the hands of Dedekind. We will see that although a ring of algebraic integers need not be a UFD, unique factorization of ideals will always hold. 7.1 Integral Extensions If E/F is a field extension and α E , then α is algebraic over F iff α is a root of a polynomial with coeﬃcients in F . We can assume if we like that the polynomial is monic, and this turns out to be crucial in generalizing the idea to ring extensions.
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