## An Introduction to Algebraic Number Theory by Takashi Ono (auth.)

By Takashi Ono (auth.)

Similar number theory books

Number Theory IV: Transcendental Numbers

This booklet is a survey of crucial instructions of study in transcendental quantity idea - the idea of irrationality, transcendence, and algebraic independence of varied numbers. there's a specified emphasis at the transcendence houses of the values of certain services. The booklet comprises few whole proofs, yet particularly provides conceptual discussions of the imperative principles in the back of the proofs.

Multiplicative Number Theory

The recent version of this thorough exam of the distribution of best numbers in mathematics progressions deals many revisions and corrections in addition to a brand new part recounting fresh works within the box. The booklet covers many classical effects, together with the Dirichlet theorem at the lifestyles of top numbers in arithmetical progressions and the theory of Siegel.

Beilinson's Conjectures on Special Values of L-Functions

Beilinsons Conjectures on specified Values of L-Functions offers with Alexander Beilinsons conjectures on specific values of L-functions. subject matters coated diversity from Pierre Delignes conjecture on serious values of L-functions to the Deligne-Beilinson cohomology, besides the Beilinson conjecture for algebraic quantity fields and Riemann-Roch theorem.

Extra info for An Introduction to Algebraic Number Theory

Example text

27 is certainly useful to determine whether the equation x 2 == a (p) has a solution or not. , the algebraic number theory. 20 already hint at this. 45. Let q be a prime such that q == 1 (8) and a be an integer such that p2 l' a for any prime p and that X4 == a (q) has no solution in 7L. Prove that the equation X4 - qy4 = az 2 has no integral solution other than x = y = z = o. Also find a pair {q, a} which satisfies the above condition. 2 Basic Concepts of Algebraic Number Fields In Chapter 2, we shall extend the results of Chapter 1 (number theory in 10) to the case of an algebraic number field K.

A/ are relatively prime ¢::>the equation AIxI + ... + A/x/ = 1 has a solution Xi E K[X], 1:5 i :51. 4*. Let A, B, C be polynomials in K[ X] such that (A, B) = 1. Then A I BC=>A I c. 5*. Let P be irreducible. P lAB P IA or P I B. 6*. Every polynomial in K[X] can be written as a product of irreducible polynomials; it can be so written in one and only one way except for the order of factors and for multiplication by elements of KX. 20. 2* show that the ring R = Z or K[X] is "euclidean". , K = IFp = Z/pZ.

D. duces the group isomorphism R X =Rf x· .. x RT'. 36. 15 is surjective. 10. If(n, m) f in the proof of = 1 then cp(nm) = cp(n)cp(m). PROOF. 15 with 1=2. D. 16. When m = P'1! ) ... 15. Therefore the function cp is determined if we find cp(pe), e ~ 1. When e = 1, we find cp(p) = p - 1 because 7L/p7L is a field. 11. If P is a prime, we have q;(pe) = pe-l(p 1), e;::: 1. PROOF. Put G = (7L/pe+17L), G' = (7L/p e7L), and consider a homomorphism f: G~ G' defined by (f[X]p<+I) = [x]p" f is surjective, because if [X]pe E G' (hence p f x) we have f([X]pe+l) = [X]pe.