課程名稱︰代數二 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）︰2016/04/18 考試時限（分鐘）：180 試題 : In answering the following problems, please give complete arguments as much as possible. You may ask for any definition. For the first five problems, 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 five problems. You do not need to give proofs of the theorems (statements) you are using, but you MUST write down complete statements of the theorems which your arguments are based. For the last problem, which is a Theorem from the Textbook. You are requested to give direct proof, using only the definition of Tensor Product. For your reference, formulas for quartic equations are give here: 4 3 2 4 f(x) := x + px + qx + r = Π (x - α ). i=1 i 4 3 2 2 2 2 4 3 D = 16p r - 4p q - 128p r + 144pq r - 27q + 256r . f The resolvent cubic for the equation f(x) = 0 is: 3 2 2 h(x) = x - 2px + (p - 4r)x + q . The roots of h(x) are: (α + α )(α + α ), (α + α )(α + α ), (α + 1 2 3 4 1 3 2 4 1 α )(α + α ). 4 2 3 ＿1/2 1. Prove that |Q((2 + √2) ) is a cyclic Galois extension of |Q. 2. Let p ≠ 3, 5 be a prime number. Compute the Galois group of the polynomial 4 x + px + p over |Q. 3. Let K be a finite extension of |Q. Let f(x) ∈ K[x] be a separable ＿ polynomial. For each embedding σ of K into the field of algebraic numbers |Q, σ let f (x) be the polynomial obtained from f(x) by applying σ to the coefficients of f(x). Then define the norm of f(x) to be σ N(f)(x) := Π f (x), σ ＿ where σ runs through all the embeddings of K into |Q. (1) Show that N(f)(x) ∈ |Q[x]. (2) Suppose that both f(x) and N(f)(x) are separable polynomials. Let N(f)(x) = n Π N (x) be the factorization into irreducible polynomials in |Q[x]. Prove that i=1 i n f(x) = Π gcd(f(x), N (x)), i=1 i is a factorization of f(x) into irreducible factors inside K[x]. ＿ ＿ ＿ 4. Find a primitive generator for the field |Q(√2, √3, √5) over |Q. In other ＿ ＿ ＿ words, give an algebraic number α satisfying |Q(α) = |Q(√2, √3, √5). ＿ ＿ 1/2 5. Let α := ((2 + √2)(3 + √3)) , and E := |Q(α). ＿ ＿ ＿ ＿ (1) Show that a := (2 + √2)(3 + √3) is not a square in F := |Q(√2, √3). ＿ ＿ (Hint: Take norm N ＿ from F to |Q(√2). This will contradicts √6 is F/|Q(√2) ＿ not an element of |Q(√2).) (2) Prove [E: Q] = 8, and the roots of the minimal polynomial of α over |Q are given by ＿ ＿ 1/2 ±((2 ±√2)(3 ±√3)) . ＿ ＿ 1/2 (3) Let β := ((2 - √2)(3 + √3)) . Show that αβ ∈ F so that β ∈ E. Show furthermore that E is Galois over |Q. (4) Let σ ∈ Gal(E/|Q) be the automorphism sending α to β. Show that σ is an element of order 4 in Gal(E/|Q). ＿ ＿ 1/2 (5) Let τ be the automorphism defined by τ(α) := ((2 + √2)(3 - √3)) . Show that τ is also of order 4 in Gal(E/|Q). Show that τ and σ generate 2 2 3 Gal(E/|Q), σ = τ , and στ = τσ . In other words Gal(E/|Q) is the quaternion group of order 8. 6. Let R be a commutative ring (with multiplicative identity 1). let M, M', N, N' be R-modules. Give R-module homomorphisms φ: M → M', ψ: N → N'. (1) Show that there exists a unique R-module homomorphism from M \otimes N to R M' \otimes N' sending m \otimes n to φ(m) \otimes ψ(n) for all m ∈ M and n R ∈ N. This homomorphism is denoted by φ \otimes ψ in the following discussion. (2) If λ: M' → M", and μ: N' → N" are also R-module homomorphisms, prove that (λ \otimes μ)。(φ \otimes ψ) = (λ。φ) \otimes (μ。ψ). -- 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。 --

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