In this question we only consider subfields of R. Recall that a field extension B/F is...
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In this question we only consider subfields of R. Recall that a field extension B/F is a pure extension of type m = 2 if B = F(a), where a F but a F. A tower of number fields F = Bo C B C ... C B is a radical tower were each Bi+1/B; is a pure extension of type 2 and F Q. Let S denote the union of all towers of number fields. The numbers in S are called surds. So a surd is a real num- ber that is in some tower of number fields. (a) Prove that S is a field. Note that QCSCR. (b) Let B/Q be a field extension. Let p(x) B[x] be a polynomial with coefficients in B. Suppose a + bre B() is a root of p(x), where r E B but r B. Show that a br is also a root of p(x). Hint: apply the 'conjugation' to the equation p(a+br) = 0. Here o denotes the automorphsm of B() fixing B pointwise, determined by (a+br)=a-br. (c) Let p(x) be a cubic polynomial with rational coefficients. Show that the sum of the roots of p(x) is a rational number. (d) Using the notation of part (b), assume p(x) = B[x] is a cubic polynomial. Prove that if a + br E B(r) is a root of p(x) then p(x) also has a root in B. Hint: parts (b), (c). (e) Use induction to prove that if a cubic polynomial p(x) Q[x] has a surd root, it must have a rational root. (f) Let denote a 20 degree angle. Show that (2 cos 0)3-3(2 cos 0)- 1=0. (g) Show that x0 = 2 cos 0 is not a surd. Hint: by part (f), we know xo is a root of the cubic polynomial p(x) = x-3x-1 e Q[x]. So if zo where a surd then p(x) would have a rational root. Show that p(x) has no rational roots (can use rational root the- orem). Remark: showing that cos 20 is not a surd is the main part of proving that an angle of 60 cannot be trisected using a straightedge and a compass. In this question we only consider subfields of R. Recall that a field extension B/F is a pure extension of type m = 2 if B = F(a), where a F but a F. A tower of number fields F = Bo C B C ... C B is a radical tower were each Bi+1/B; is a pure extension of type 2 and F Q. Let S denote the union of all towers of number fields. The numbers in S are called surds. So a surd is a real num- ber that is in some tower of number fields. (a) Prove that S is a field. Note that QCSCR. (b) Let B/Q be a field extension. Let p(x) B[x] be a polynomial with coefficients in B. Suppose a + bre B() is a root of p(x), where r E B but r B. Show that a br is also a root of p(x). Hint: apply the 'conjugation' to the equation p(a+br) = 0. Here o denotes the automorphsm of B() fixing B pointwise, determined by (a+br)=a-br. (c) Let p(x) be a cubic polynomial with rational coefficients. Show that the sum of the roots of p(x) is a rational number. (d) Using the notation of part (b), assume p(x) = B[x] is a cubic polynomial. Prove that if a + br E B(r) is a root of p(x) then p(x) also has a root in B. Hint: parts (b), (c). (e) Use induction to prove that if a cubic polynomial p(x) Q[x] has a surd root, it must have a rational root. (f) Let denote a 20 degree angle. Show that (2 cos 0)3-3(2 cos 0)- 1=0. (g) Show that x0 = 2 cos 0 is not a surd. Hint: by part (f), we know xo is a root of the cubic polynomial p(x) = x-3x-1 e Q[x]. So if zo where a surd then p(x) would have a rational root. Show that p(x) has no rational roots (can use rational root the- orem). Remark: showing that cos 20 is not a surd is the main part of proving that an angle of 60 cannot be trisected using a straightedge and a compass.
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