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.