国台湾师范大学九十三学博士班考试入学招生试题

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(a) Give an example that N K / F is not surjective. (b) Suppose that F is a finite field. Show that G is cyclic by describing G explicitly. (c) Suppose that F is a finite field. Show that N K / F is surjective.國立臺灣師範源自學九十三學年度博士班考試入學招生試題
代數科試題 （數學研究所用，本試題共 2 頁第 1 頁）
注意： 依次序作答，只要標明題號，不必抄題。 答案必須寫在答案卷上，否則不予計分。
總分 100 分，每題 20 分 Notations： Z : the ring of integers Q : the field of rational numbers C : the field of complex numbers 1. Let G be a finite group and let H be a subgroup of G. Recall that the normalizer of H in G is defined by N(H)＝{a ∈ G | aHa −1 ＝H}. (a) Show that N(H) is the largest subgroup of G in which H is normal. (b) Show that the number of distinct conjugates xHx −1 of H in G is equal to [G : N(H)]. (c) Suppose that G is of order 48. Show that G is not simple. 2. Consider the permutation group S 4 and the alternating group A 4 . (a) Describe all the conjugacy classes of S 4 and write down its class equation. (b) Show that there exists a normal subgroup K of A 4 which is isomorphic to the Klein four group. (c) Show that S 4 is solvable. (d) Does A 4 have a subgroup of order 6 ? 3. An element α of C is said to be integral over Z if there exists a nonzero monic polynomial f(x) over Z such that f( α )＝0. (a) Show that the following statements are equivalent : (i) α is integral over Z. (ii) There exists a nonzero finitely generated Z-submodule M of C such that α M ⊂ M. (b) Determine all the elements of the quadratic field Q( − 3 ) which are integral over Z. 4. For a finite Galois extension K/F with Galois group G , the norm map N K / F : K → F is defined by N K / F (a)＝ ∏ σ (a) , where a ∈ K.
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國立臺灣師範大學九十三學年度博士班考試入學招生試題
代數科試題 （數學研究所用，本試題共 2 頁第 2 頁）
注意： 依次序作答，只要標明題號，不必抄題。 答案必須寫在答案卷上，否則不予計分。
5. Determine the Galois groups of the following polynomials over Q. (a) x 3 －3x＋1 (b) x 4 －2 (c) x 5 －4x＋2 (Note : If you use certain theorems you knew, please write down the statements of the theorems you used.)