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）︰2016/05/23 考試時限（分鐘）：180 試題 : 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.) -- 第01話 似乎在課堂上聽過的樣子 第02話 那真是太令人絕望了 第03話 已經沒什麼好期望了 第04話 被當、21都是存在的 第05話 怎麼可能會all pass 第06話 這考卷絕對有問題啊 第07話 你能面對真正的分數嗎 第08話 我，真是個笨蛋 第09話 這樣成績，教授絕不會讓我過的 第10話 再也不依靠考古題 第11話 最後留下的補考 第12話 我最愛的學分 --

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