课程名称︰代数导论二 课程性质︰必修 课程教师︰陈其诚 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2016/6/2 考试时限（分钟）：120 试题 : Let F_q denote a field of order q. Part I (30 points) True or false. Either prove the assertion or disprove it by a counter-example (6 point each): (1) The polynomial x^8-x splits in F_8[x]. (2) The Galois group Gal( F_125 / F_5 ) is trivial. (3) The Galois group Gal( Q ( 5^(1/3) ) / Q ) is trivial. (4) Let K be the splitting field of x^3-2x+1. The Galois group Gal( K / Q ) is isomorphic to S_3. (5) Let n be an integer. The real number n^(1/3) is contained in Q( 7^(1/3) ) if and only if n = 7m^3 or n = 49m^3, m∈Z. (6) The fixed field of the subgroup <τ> of Gal ( C(t) / C ) such that τ(t) = -t is C(t^2). Part II (40 points). For each of the following problems, give accordingly a short proof or an example (8 point each): (1) There are totally 8 monic irreducible degree 3 polynomials in F_3[x]. (2) Determine the minimal polynomial of √5 + √3 over Q. (3) Determine the Galois group of Q(ζ)/Q, where ζ = e^(2πi/11). (4) Find a Galois extension with Galois group isomorphic to Z/nZ, for any n. (5) Find a Galois extension with Galois group isomorphic to D_4. (6) Find a Galois extension with Galois group isomorphic to S_n, for any n. Part III (30 points). Give a complete proof of the following (10 points each): (1) Let L/K be a Galois extension and let α∈L. Denote B = {σ(α) | σ∈Gal(L/K) }. Show that the polynomial Π (x-β) has coefficients in K β∈B and is actually the minimal polynomial of α over K. (2) Determine the Galois group Gal( Q(√5,√3) / Q ). (3) Show that the splitting field of x^3+x+1 over Q contains no root of x^3+3x+1. --

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