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Mathematics 666 Homework (due Jan. 29) A. Hulpke 1) Consider the following properties for commutative rings with 1: Euclidean, Integral Domain, Noetherian, Principal Ideal Domain, Unique Factorization Domain. Indicate (e.g. by a Venn-type diagram) which properties imply which others (you do not need to prove this), and give examples of rings that show that thes properties are distinct, i.e. that fulfil one property but not another. (Indicate why these examples have the desired properties). 2) a) Let α = 8 ⋅ sin(6○ ). One can show that α is an algebraic number of degree 4. In this problem we want to determine the minimal polynomial of α: 1. Calculate α numerically (to 20 or more digits – it seems that with less accuracy there are problems) 2. For N = 1015 , calculate the integers N, ⟨N α⟩, ⟨N α 2 ⟩,. . . , ⟨N α 5 ⟩. 3. Using the LLL-algorithm, find a Z-linear combination of these numbers that is small, has small coefficients, and in which the coefficient of x 5 is zero. From these coefficients determine the minimal polynomial of α. b) (voluntary) Express α in terms of radicals. √ √ 3) Let A = Z[ 5] = {a + b −5 ∣ a, b ∈ Z}. √ 2 2 a) Determine the units √of A. Hint: Consider the norm map: a + b −5 → a + 5b . b) Show that 2, 3,√1 ± −5 are irreducible in A. √ √ c) Let J = (2, 1 + −5), K = (3, 1 + −5), R = (3, 1 − −5). Show that P, Q, R are maximal ideal in A, containing (3), but not principal √ ideals. √ d) Show that J 2 = (2), J ⋅ K = (1 + −5), J ⋅ R = (1 − 5), K ⋅ R = (3) and J 2 ⋅ K ⋅ R = (6).