By Diamond H. (ed.)

ISBN-10: 0821814249

ISBN-13: 9780821814246

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Therefore F(0'2) ~ F(o:), as required. 2 - - - - - - - - 1. What does the definition of F (Q') say in each of the following cases? (i) degfo , F) (ii) degf o , F) = 1, = 2, (iii) degfo , F) = 3. 2. (a) What do es the definiti on of F( Q') tell you in the sp ecial case where F = Q and Q' = V3 ? 4 t ell you in this sp ecial case? ,13). 3. What is the degree of V2 over R? Whi ch familiar field is R(V2 )? 4. What is the degree of i over R? Which familiar field is R(i)? 5. (a) Stat e why 1 + V2 E Q ( V2) and why V2 E Q (l + V2).

3). (a) Use the fact that any m + 1 vectors in a vect or space of dimension m arc linearl y dep end en t , and your ans wer to Exer cise 8(i), t o show that {I , (3, (32, (33,(34} is linearly dependent over Q.

1), all E F(a). If we multiply both sides of (1) by a we see that (2) Now a, ... , a,,-I are all in IF(a) (by definition) and we have just shown that an E IF(a). Thus, because IF(a) is a vector subspace of C, it follows from (2) that a ll + 1 E F(a) . Now multiply both sides of (1) by a 2 and then proceed in the same way, thereby showing that a ll+ 2 E IF(a). {. all belong to IF(a). •• • • Thus, in our definition of F(a) we have, in some sense, included "enough" powers of a. Have we included "too many"?

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Analytic Number Theory by Diamond H. (ed.)


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