课程名称︰代数二
课程性质︰数学系大二必修
课程教师︰于靖
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/05/23
考试时限（分钟）：180
试题 :
This is an Open Book Examination. You may take the textbook by Dummit-Foote
as a reference during the examination. In answering the following problems,
please give complete arguments as much as possible. You may ask for any
definition. You may use freely any Theorem already proved (or Lemmas,
Propositions) from the Course Lectures, or previous courses on Linear Algebra.
Previous Exercises assignments are also allowed to use in doing these problems.
You do not need to give proofs of the theorems (statements) you are using, but
you MUST write down the complete stastements of the theorems on which your
arguments are based.
5
1. Compute the Galois group of x + 2 ∈ |Q[x]. (Hint: Use cyclotomic
extensions.)
2 3
2. Let I := (zw - y , xy - z ) ⊂ |C[x, y, z, w]. Compute the dimension of the
4
affine algebraic set Ｚ(I) inside |C . (Hint: You are encouraged to use Gröbner
basis.)
3. Let α be a root of irreducible polynomial p(x) ∈ F[x], F is a fixed field.
Let β := f(α)/g(α), with f(x), g(x) ∈ F[x] and g(α) ≠ 0.
(1) Show there exists a, b ∈ F[x] s.t. ag + bp = 1. Also there exists h ∈
F[x] with β = h(α).
(2) Show that the ideals (p, y - h) and (p, gy - f) are equal in F[x, y]. Let
G be the reduced Gröbner basis for this ideal, w.r.t. the lexicographic
ordering x > y. Prove that G ∩ F[y] gives the irreducible minimal polynomial
for β over F.
(3) Find the minimal polynomial over |Q of
1/3 1/3 1/3 1/3
(3 - 2 + 4 ) / (1 + 3(2 ) - 3(4 )).
2
4. Let a ∈ |Q. Let I ⊂ |Q[x, y] be the ideal (x) ∩ (x , y - ax). Prove that
a
all these ideals I are equal. Explain that this is an ideal which has
a
infinitely many distinct minimal primary decompositions. Determine the isloated
and embedded prime ideals of this ideal.
2 3 2 4 4 2 5
5. Show that P := (xz - w , xw - y , y z - w ) ⊂ |Q[x, y, z, w] is a prime
ideal. (Hint: Use Gröbner basis and localizations.)
6. Let R be a commutative Noetherian ring. Let M, N be finitely generated R-
modules. There is a natural R-module structure on the abelian group Hom (M, N)
R
given by
(rf)(m) := rf(m), ∀f ∈ Hom (M, N), r ∈ R, m ∈ M.
R
Let D ⊂ R be a fixed multiplicative subset containing 1 .
R
(1) Show that there are finitely generated free R-modules F , F and exact
0 1
sequence:
F → F → M → 0.
1 0
Furthermore show that this sequence induces exact sequences:
-1 -1 -1
D F → D F → D M → 0,
1 0
0 → Hom (M, N) → Hom (F , N) → Hom (F , N).
R R 0 R 1
-1
(2) Given f ∈ Hom (M, N), let D f be the map sending m/d to f(m)/d, which
R
-1 -1
is an element in Hom (D M, D N). Verify that canonically
D^{-1}R
-1 -1 -1
D Hom (M, N) \cong Hom (D M, D N).
R D^{-1}R
You may prove the special case that M is free R-module first. (Hint: Use the
universal property for localization and "morphism" between exact sequences.)