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Solutions to problem 4, 5, and 6 from homework 5b of the abstract algebra ii course (math 252). The problems involve proving equalities of field extensions, finding irreducible polynomials, and showing that the 'multiplication by √d' map is a linear transformation. Problem 7 is a hint for constructing regular 5-gons using trigonometric identities or primitive roots of unity.
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Problem 4 (DF 13.2.7). Prove that Q(
3). [Hint: Consider (
2 .] Conclude
that [Q(
Problem 5 (DF 13.2.14). Prove that if [F (α) : F ] is odd then F (α) = F (α^2 ).
Problem 6 (DF 13.2.21). Let D ∈ Z be squarefree and let K = Q(
D). Let α = a + b
(a) Show that the “multiplication by α” map
φ : K → K
β 7 → φ(β) = αβ
is a linear transformation (of vector spaces over Q).
(b) Compute the matrix of φ on the basis 1,
D of K.
Problem 7 (sorta DF 13.3.5).
(a) Show that α = 2 cos(2π/5) satisfies the equation x^2 + x − 1 = 0. [Hint: Use a trigonometric identity or the fact that α = ζ 5 + 1/ζ 5 where ζ 5 = exp(2πi/5) = cos(2π/5) + i sin(2π/5) is a primitive fifth root of unity.] (b) Conclude that the regular 5-gon is constructible by straightedge and compass.
Date: 24 March 2008; due 4 April 2008. 1