A Brief Introduction to Classical and Adelic Algebraic by Stein W. PDF
By Stein W.
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Extra info for A Brief Introduction to Classical and Adelic Algebraic Number Theory
Here’s a suggestive picture: (5, 2 + 4a + a2 ) ✏ ✑ ✏ ✑ ✟ ✠ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ 2OK ✡ ☛ (5, 3 + a)2 (5, 2 + a) (0) ✁ ✁ (0) 2Z 5Z 3Z 7Z 11Z ✂ ✂ ✄ ✂ ✂ ✄ ✝ ✞ ☞ ✌ ☎ ✆ ✝ ✞ ☞ ✌ ☎ ✆ Diagram of Spec(OK ) → Spec(Z) 52 CHAPTER 8. 1 A Method for Factoring that Often Works Suppose a ∈ OK is such that K = Q(a), and let g(x) be the minimal polynomial of a. ) // Spec(OK ) Spec(Fp [x]/(g ei i )) Spec(Fp ) // Spec(Z[a]) // Spec(Z) where g = i g ei i is the factorization of the image of g in Fp [x].
Using linear algebra over the finite field Fp , we can quickly compute a basis for I/pO. ) 3. [Compute quotient by radical] Compute an Fp basis for A = O/I = (O/pO)/(I/pO). The second equality comes from the fact that pO ⊂ I, which is clear by definition. Note that O/pO ∼ = O ⊗ Fp is obtained by simply reducing the basis w1 , . . , wn modulo p. 4. [Decompose quotient] The ring A is a finite Artin ring with no nilpotents, so it decomposes as a product A ∼ = Fp [x]/gi (x) of fields. 5]. 5. [Compute the maximal ideals over p] Each maximal ideal pi lying over p is the kernel of O → A → Fp [x]/gi (x).
48 CHAPTER 7. COMPUTING Chapter 8 Factoring Primes First we will learn how, if p ∈ Z is a prime and OK is the ring of integers of a number field, to write pOK as a product of primes of OK . Then I will sketch the main results and definitions that we will study in detail during the next few chapters. We will cover discriminants and norms of ideals, define the class group of OK and prove that it is finite and computable, and define the group of units of OK , determine its structure, and prove that it is also computable.
A Brief Introduction to Classical and Adelic Algebraic Number Theory by Stein W.