Download e-book for iPad: Algebraic and Geometric Topology, Part 2 by Milgram R. (ed.)

By Milgram R. (ed.)

ISBN-10: 0821814338

ISBN-13: 9780821814338

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Additional resources for Algebraic and Geometric Topology, Part 2

Sample text

Proof. I f the field F has characteristic n ^ 0, then n is a The proof is left as an exercise. 9(vi). 6 Let F be an arbitrary field. Form the plane = as follows. Let {[x,y,z]' x,y,z ^ F a n d x = y = z = 0 is not true} where [x,y,z] = [ x \y \ z ' ] if and only if there exists r El F, r 7^ 0, such that x = rx \ y = ry’, z = rz'• Let oSf = {{a,b,c): a,b,c E F and a = b = c = 0 is not true} where (a,b,c) = ( a \ b \ c ' ) if and only if there exists ^ E F, 5 7^ 0, such that a = s a \ b = s b \ c = sc'.

Also, “afc” will be used as a shorthand notation to denote a • b. Many different kinds of algebras will be introduced in this book, and to preserve distinctions it will be necessary to use several different symbols to indicate addition and multiplication. For example, © and S will also be used for addition and °, 0 , ©, □ , and © will be used for multiplication. The multidot (•) is the only multiplication symbol that will be omitted in multiplication. 2 i. The real numbers, with ordinary addition and multiplication, form a field.

This field is denoted by R. ii. The complex numbers, with ordinary addition and multiplication, form a field. This field is denoted by C. iii. The rational numbers, with the ordinary operations, form a field. This field is denoted by Q. 39 PLANES DEFINED BY FIELDS iv. Let p be any prime number and let the set of elements of 5 = {[0],[1], . . /::/: G J} and where J denotes the set of all integers. Define addition, ©, and multipli­ cation, o, as follows: [/] © [j] = [i + j], [/] ° [j] = [/ • j].

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Algebraic and Geometric Topology, Part 2 by Milgram R. (ed.)


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