By Paul J. McCarthy

ISBN-10: 0486666514

ISBN-13: 9780486666518

*Math.*studies. Over two hundred routines. Bibliography.

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Show that Kis a Galois extension of GF(pn). Let the automorphism a of K be given by a(a) = a1~'' for all a E K. Show that G(K/GF(pn)) is cyclic and generated by a. 68 Exercises Galois theory 8. Let k be a finite field and K a finite extension of k. Show that every nonzero element of k is the norm of exactly (K*: k*) elements of K*. Show by example that this is not true, in general, when k is an infinite field. (Hint. ) 9. A field k is called a quasi-finite field if k is perfect and if it has, in any algebraic closure, exactly one extension of degree n for each integer n > 0.

Let n > 4 and (h,n) = cp(n)/4 if (n,8) = 4 cp(n)/2 is {n,8) > 4. cp(n) if (n,8) <4 cp(n)/2 if (n,8) = 4 cp(n)/4 if (n,8) 1. Show that Section 4 19. The results of this exercise are needed in the proof of Proposition 4 of Section 4. A polynomial f(x) E J[x] is called pri1nitive if the g. c. d. of its coefficients is one. + F 12(x) = x 4 - x 2 + 1, and in general, for a prime p, that I I 69 [Q(tan 21Th/n): Q] = > 4. 70 71 Exercises Galois theory 28. Find a basis of Kn over Q and determine the discriminant of this basis.

See ~xercise 4). Suppose th~t, for i = 1, ... • , an as bi ts of zh ... , Zn. Define the mapping a from F[y1, ... , Ynl into F[bh ... , bn] by a(g{yh ... , Yn)) = g(b1 , ••• , bn): (] is clearly a homomorphism onto F[b1 , ••• , bn1· Suppose a(g(y1 , • •• , Yn)) = 0. ' ... , bn) = 0 and if we replace each bi by its expression blz1, ... , Zn) tn terms of zh ... , Zn we have g(b1(zb ... , z~), ... , bn{zh ... , zn)) = 0. Since zh ... , Zn are independent indeterminates, the result of substituting any elements of any field containing F for zh ...

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