Algebra Homework 5 Due by the start of class on Wednesday Nov. 5 Problem 1: 1. Draw steps for how to bisect an angle using a straightedge and compass. You shouldn’t need more than two or three pictures, and you don’t have to “prove” that your method actually bisects the angle. A convincing picture is enough. 2. Prove that if a regular n-gon is constructible, then a regular 2n-gon is also constructible. Your “proof” should consist of a precise description of how to construct the 2n-gon. Problem 2: Prove that π 2 − 1 is algebraic over Q(π 3 ). √ √ √ √ Problem 3: Prove that Q( 3, 3 3, 4 3, 5 3, . . . ) is an algebraic field extension of Q, but that it is not finite. Problem 4: Let F be a field and E be the splitting field for a polynomial f (x) ∈ F [x], where deg(f (x)) = n. 1. Show that [E : F ] ≤ n!. (the exclamation point denotes factorial). 2. Let p be any prime number bigger than n. Show that p does not divide [E : F ]. Problem 5: Describe the splitting field for x4 − x2 − 2 ∈ Z3 [x].
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