課程名稱︰代數一
課程性質︰數學系大二必修
課程教師︰于靖
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/01/11
考試時限（分鐘）：180
試題 :
1. Let m, n, and l be positive integers, ι: Ｚ/(m) ×Ｚ/(n) ×Ｚ/(l) →
Ｚ/(gcd(m, n, l)) be the map defined by:
(a mod m, b mod n, c mod l) ├→ abc mod gcd(m, n, l)
(ⅰ) Verify that this map ι, as a function of three variables on its domain,
is a Ｚ-trilinear map, i.e., Ｚ-linear in each of the three variables.
(ⅱ) Show that given any abelian group A, and Ｚ-trilinear map φ: Ｚ/(m) ×
Ｚ/(n) ×Ｚ/(l) → A, there is a unique Ｚ-homomorphism Φ: Ｚ/(gcd(m,
n, l)) → A satisfying Φ。ι = φ.
2. (ⅰ) Let R be a commutative ring with multiplicative identity 1. Given R-
modules M, N, L. Show that the set of all R-bilinear maps from M ×N to
2
L is naturally also an R-module (denoted below by L (M, N; L)). Moreover
R
show that this R-module is canonically isomorphic to Hom (M, Hom (N,
R R
L)).
(ⅱ) Let m, n, and l be positive integers. Determine the following Ｚ-module
up to isomorphism:
2
L (Ｚ/(m), Ｚ/(n); Ｚ/(l)).
Ｚ
3. Let A be the following matrix with entries from |R:
┌ 1 2 -4 4 ┐
│ │
│ 2 -1 4 -8 │
│ │
│ 1 0 1 -2 │
│ │
└ 0 1 -2 3 ┘
t
Let A be its transpose. Find an invertible matrix P ∈ M (|R) such that
4
-1 t
PAP = A .
4. Let F be a field. Suppose N ∈ M (F) is a square matrix which is nilpotent,
n
k n
i.e., satisfying N = 0 for some positive integer k. Prove that N = 0 must
hold.
5. Let F be a field, F[x , ..., x ] be the polynomial ring in n variables. A
1 n
polynomial f ∈ F[x , ..., x ] is said to be symmetric if f(x , ..., x ) =
1 n 1 n
f(x , ..., x ) holds for all permutations σ ∈ S . The elementary
σ(1) σ(n) n
symmetric polynomials s , ..., s are defined by:
1 n
s := x + x + ... + x ,
1 1 2 n
s := x x + x x + ... + x x + x x + ... + x x ,
2 1 2 1 3 2 3 2 4 n-1 n
.
.
.
s := x x ...x .
n 1 2 n
Fix monomial order so that x > x > ... > x .
1 2 n
(ⅰ) Given nonzero symmetric f ∈ F[x , ..., x ]. Write its leading term
1 n
α α_1 α_n
LT(f) := ax , with a ∈ F and x = x ... x . Show that the
1 n
multidegree α = (α , ..., α ) satisfies α ≧ α ≧ ... ≧ α .
1 n 1 2 n
Moreover let:
α_1 - α_2 α_2 - α_3 α_{n-1} - α_n α_n
h := s s ... s s
1 2 n-1 n
Verify that the multidegree of f - ah is less than the multidegree of f.
2 2 2 2 2 2
(ⅱ) Let n = 3. Write the symmetric polynomial (x + x )(x + x )(x + x ) as
1 2 2 3 3 1
a polynomial in the elementary symmetric polynomials s , s , s .
1 2 3
6. Let M be a free Ｚ-module of rank n > 0. Let f: M → M be a given Ｚ-linear
homomorphism in Hom (M, M). Let B = {v , ..., v } be an (ordered) basis for
Ｚ 1 n
M. For 1 ≦ j ≦ n, write
n
f(v ) = Σ a v ,
j i=1 ij i
with a ∈ Ｚ. This gives n ×n matrix A := (a ).
ij ij
(ⅰ) Prove that M/f(M) is a torsion Ｚ-module if and only if det(A) ≠ 0.
(ⅱ) If det(A) ≠ 0, show that the index of the subgroup f(M) inside the
abelian group M is equal to |det(A)|.