臺灣大學數學系

臺灣大學數學系

99 學年度下學期博士班資格考試題

科目:代數

201 1. 02.25

(1) (15 %) Let k be a fie祠， and k[[x)] be the set of all formal power series L:立。的計，

with coefficientsωε k. Define operations on k[[xJJ by

Laixi

+

Lbixi = 芝(何 +b拭

i=O i=O i=O

C由 C自 C自.

(Lai xi ) . (2二 b叫)=芝(L αjbi_j)x'

i=O i=O i=O j=O

Show that k[[x)] is a principal ideal domain under these operations. Furthermore show that there is only one maximal ideal inside this ring.

(2) (15 %)Let Matn(lQ) be the ring consisting of n x n square matrices with entries from IQ. Prove that any automorphism of this ring must be of the form M 0-+ AMA一 1 ， for some invertible matrix A εMatn(IQ).

(3) (15 %) Let M be any given n x n matrix having entries from field k

,

with char­

'acteristic polynomial P(x) ε k[xJ. Give a proof of the following identity (Cayley­

Hamilton) :

P(M)

=

O.

(4) (15 %) Let 1F32 be the finite fìeld with 32 elements, G := GL32(1F32) is the group of invertible 32 x 32 matrices having entries from lF32. Show that the subgroup U of G consisting of upper trangular matrices having its diagonal entries all equal to 1 is a Sylow 2-subgroup of G. Then prove that any finite group of order 32 is isomorphic to a subgroup of this matrix group U (you may use Sylow's theorems to prove this sta指ment).

(5) (10%) Use Galois theory to compute the Galois group of the polynomial x⁵ - 5 over IQ.

(6) (15 %) Let P ba a given maximal ideal in the ring Z[AJ. Show first that

pnz =

(P), where p is a prime number. Denote by P the set of complex conjugates of elements in P. Then veri秒 that if 吭，向 are two maximal ideals in Z[V=可 satisfyi呀 Pi

n z =

(P), for i

=

1,2, then either P1

=

P2 , or P1

=

P2 (you may apply Chinese remainder theorem to the ring Z[A]).

(7) (15 %) Let V := Cⁿ be the 點dimensional complex vector space equipped with the hermitian form

(ω) ;= 芝叭， U

,

^{V E Cn}

‘

⁼¹

Recall that a linear transformation T : V

•

^V is said to be selfcadjoint if it satisfies:

(T(包)， υ)

=

(包， T(v)) ， Vu ,v E V.

Given a set S of self-adjoint transformations on V which are mutually commuting, i.e. T₁ ⁰ T2 ⁼ T2 ⁰ T1' if T ₁,T2 E S. Prove that this sp缸e V must have a basis

consi的ing of elements which are eigenvectors for all transformations in S. 1