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)|. -- 移居二次元(|R^2)的注意事項： 3. 如果你在從事random walk，往上下左右的 1. connectedness不保證pathwise connec- 的機率都是1/4，則你能回家的機率是1。 tedness。可能你跟你的幼馴染住很近， 4. 下面這個PDE是二次元上的波方程式 卻永遠沒辦法到她家。 http://i.imgur.com/2H9HllP.png 2. ODE的C^1 autonomous system不會出現 它的解不滿足Huygens' principle，因此 chaos，在預測事情上比較方便。 講話時會聽到自己的回音，很不方便。 --

※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 140.112.212.7 ※ 文章網址: https://webptt.com/m.aspx?n=bbs/NTU-Exam/M.1452537608.A.47B.html ※ 編輯: xavier13540 (140.112.212.7), 01/12/2016 02:53:51