MATH 252: ABSTRACT ALGEBRA II
HOMEWORK #5B
Problem 4 (DF 13.2.7). Prove that Q(√2 + √3) = Q(√2,√3). [Hint: Consider (√2 + √3)2.] Conclude
that [Q(√2 + √3) : Q] = 4. Find an irreducible polynomial over Qsatisfied by √2 + √3.
Problem 5 (DF 13.2.14). Prove that if [F(α) : F] is odd then F(α) = F(α2).
Problem 6 (DF 13.2.21). Let D∈Zbe squarefree and let K=Q(√D). Let α=a+b√D∈K.
(a) Show that the “multiplication by α” map
φ:K→K
β7→ φ(β) = αβ
is a linear transformation (of vector spaces over Q).
(b) Compute the matrix of φon the basis 1,√Dof K.
Problem 7 (sorta DF 13.3.5).
(a) Show that α= 2 cos(2π /5) satisfies the equation x2+x−1 = 0. [Hint: Use a trigonometric identity
or the fact that α=ζ5+ 1/ζ5where ζ5= exp(2πi/5) = cos(2π/5) + isin(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.
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