Download Algebra 2 TD Spring 2017 Exercise sheet 10 †66. [Ruler and

Algebra 2
Spring 2017
Exercise sheet 10
TD
† 66.
[Ruler and compass] Which of the following real numbers are constructible?
q
√
√ √
√
17
3
2 + 3; roots of x3 −7x−6 = 0;
11;
2+ 4
17/3;
√ √ √
√
67. Show that the field Q( 3 2, 3 3, 3 5) ⊂ R does not contain 5 2.
68. Prove that for every prime p the polynomial xp − x + 1 has no roots in Fp .
69. Prove that a complex number is algebraic if and only if its real and imaginary part
are algebraic.
70. How many solutions does each of the equations x3 = 1, x5 = 1 and x7 = 1 have in
the field F16 ? [This is a question about orders of elements in the group F×
16 .]
71. Suppose K is a field for which every polynomial f ∈ K[x] has all its roots. (Such
fields are called algebraically closed, and two examples are the field C of complex
numbers and the field Q̄ of algebraic numbers.) Prove that K is infinite.
72. Let K be a field. For a polynomial f = an xn + an−1 xn−1 . . . + a1 x + a0 ∈ K[x] define
formally define f 0 ∈ K[x] by the formula
f 0 = nan xn−1 + (n−1)an−1 xn−2 + . . . a1 .
(1) Show that (f + g)0 = f 0 + g 0 and (f g)0 = f 0 g + f g 0 for all f, g ∈ K[x].
(2) Suppose f = g 2 h for some f, g, h ∈ K[x] and degree(g) > 0. (For example
this is true if f has a multiple root, that is f = (x − α)2 h.) Prove that in
that case f and f 0 are not coprime.
73. Suppose p is a prime number, and n ≥ 1 an integer. Use the previous exercise to
n
prove that if K is a finite extension of Fp in which xp − x has all its roots, then
these roots are all distinct (no multiplicities).
∗ 74.
Show that xn + x + 3 ∈ Q[x] is irreducible for every n ≥ 1.
Exercise 66 (marked with a †) is the last assignment for this course.
Please hand it in on Thursday April 27 by 3.30pm.