By Ivan Fesenko

Creation to algebraic quantity theory

This direction (36 hours) is a comparatively uncomplicated path which calls for minimum must haves from commutative algebra for its knowing. Its first half (modules over crucial perfect domain names, Noetherian modules) follows to a undeniable volume the publication of P. Samuel "Algebraic concept of Numbers". Then integrality over earrings, algebraic extensions of fields, box isomorphisms, norms and strains are mentioned within the moment half. frequently 3rd half Dedekind jewelry, factorization in Dedekind earrings, norms of beliefs, splitting of top beliefs in box extensions, finiteness of definitely the right type crew and Dirichlet's theorem on devices are handled. The exposition occasionally makes use of equipment of presentation from the booklet of D. A. Marcus "Number Fields".

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**Extra info for Algebraic Number Theory**

**Sample text**

This way we get l units u1 , . . , ul . We claim that there are l − 1 linearly independent vectors among the images g(ui ). To verify the claim it suffices to check that the first l − 1 columns of the matrix (αi(k) ) are linearly independent. If there were not, then there would be a non-zero vector (t1 , . . , tl−1 ) such that l−1 (k ) = 0 for all 1 k l . Without loss of generality one can assume that i=1 ti αi there is i0 between 1 and l − 1 such that ti0 = 1 and ti 1 for i = i0 , 1 i l − 1.

Thus, every ideal of the ring of integers of Q( 14) is principal, CQ(√14) = {1}. √ 4. It is known that for negative square-free d the only quadratic fields Q( d) with class number 1 are the following: √ √ √ √ √ Q( −1), Q( −2), Q( −3), Q( −7), Q( −11), √ √ √ √ Q( −19), Q( −43), Q( −67), Q( −163). For d > 0 there are many more quadratic fields with class number 1. Gauss conjectured that there are infinitely many such fields, but this is still unproved. 8. Now we can state one of the greatest achivements of Kummer.

3. Corollary. Every ideal class of OF contains an ideal J with N (J) c. Proof. Given ideal class, consider an ideal I of the inverse ideal class. Let a ∈ I be as in the theorem. 3 there is an ideal J such that IJ = aOF , so (I)(J) = (aOF ) = 1 in CF . Then J belongs to the given ideal class. 4 we deduce that N (I)N (J) = N (IJ) = N (aOF ) = |NF/Q (a)| cN (I) . Thus, N (J) c. 4. Theorem. The ideal class group CF is finite. The number |CF | is called the class number of F . Proof. 2 for each prime p there are finitely many maximal ideals P lying over (p), and N (P ) = pm for m 1 .

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