This e-book is meant to function a textbook for a direction in algebraic topology initially graduate point. the most issues coated are the class of compact 2-manifolds, the basic team, masking areas, singular homology conception, and singular cohomology idea. those issues are constructed systematically, averting all unecessary definitions, terminology, and technical equipment. at any place attainable, the geometric motivation in the back of a few of the innovations is emphasised. The textual content comprises fabric from the 1st 5 chapters of the author's prior e-book, ALGEBRAIC TOPOLOGY: AN creation (GTM 56), including just about all of the now out-of- print SINGULAR HOMOLOGY concept (GTM 70). the fabric from the sooner books has been rigorously revised, corrected, and taken modern.
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Additional info for A basic course in algebraic topology
4. 3 implies that the norm and trace down to L of α is an element of OL , because the sum and product of algebraic integers is an algebraic integer. 5. 5. Let K be a number field. , QOK = K and OK is an abelian group of rank [K : Q]. Proof. 19 that QOK = K. Thus there exists a basis a1 , . . , an for K, where each ai is in OK . Suppose that as x = ni=1 ci ai ∈ OK varies over all elements of OK the denominators of the coefficients ci are not all uniformly bounded. Then subtracting off integer multiples of the ai , we see that as x = ni=1 ci ai ∈ OK varies over elements of OK with ci between 0 and 1, the denominators of the ci are also arbitrarily large.
DEDEKIND DOMAINS AND UNIQUE FACTORIZATION OF IDEALS To finish the proof that p has an inverse, we observe that d preserves the finitely generated OK -module p, and is hence in OK , a contradiction. More precisely, if b1 , . . , bn is a basis for p as a Z-module, then the action of d on p is given by a matrix with entries in Z, so the minimal polynomial of d has coefficients in Z (because d satisfies the minimal polynomial of d , by the Cayley-Hamilton theorem – here we also use that Q ⊗ p = K, since OK /p is a finite set).
As we will see, in general the problem of computing OK given K may be very hard, since it requires factoring a certain potentially large integer. 3. 17 (Order). An order in OK is any subring R of OK such that the quotient OK /R of abelian groups is finite. ) As noted above, Z[i] is the ring of integers of Q(i). For every nonzero integer n, the subring Z+niZ of Z[i] is an order. The subring Z of Z[i] is not an order, because Z does not have finite index in Z[i]. Also the subgroup 2Z + iZ of Z[i] is not an order because it is not a ring.
A basic course in algebraic topology by Massey