課程名稱︰代數導論
課程性質︰必修
課程教師：陳榮凱
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰980615
考試時限（分鐘）：150分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(1)(20 pts)
Let k be a field and R=k[x,y] be the ploynomial ring with two indeterminates.
Let p=(x) be the ideal generated by x.
(a) Show that p is a prime ideal.
(b) Show that p is not a maximal ideal.
(c)Let Rp:={f/g | f(x,y),g(x,y) in R[x,y] ,g isn't in p=(x)} be the
localization and let m = {x*h/g | h(x,y),g(x,y) in R,g isn't in p=(x)}.
show that Rp/m is automorphic to K(y).
(Hint: Consider the homomorphism Rp → K(y) by sending f(x,y)/g(x,y)
to f(0,y)/g(0,y).)
(2)(15 pts)
Let K/f be an extension of degree p, whrer p is a prime number . Show that for
all u in K-F , we have K=F[u} and deg(irr(u,F))=p.
(3)(25 pts)
Determine the Galois group of X^3-5 over Q and all the intermediate subfields.
(4)(15 pts)
Let K be a finite field of P^n elements, where p is a prime number. We regard
K as an extension over Zp.
(a)Show that the polynomial x^(p^n)-x is a separable polynomial in Zp[x].
(b)Show that every element in K satisfies x^(p^n)-x in Zp[x].
(c)Show that K is Galois over Zp.
(5)(25 pts)
Let ξ= e^(2πi/5) be the 5-th root of unity. We consider the extention Q[ξ]
over Q.
(a)(10pts) Show that irr(ξ,Q)=x^4+x^3+x^2+x+1.
(b)(5 pts) Express (2ξ+1)^(-1) in terms of a3ξ^3+a2ξ^2+a1ξ+a0
for some ai in Q.
(c)(10 pts) Solve ξby radicals. (Hint: consider the intermediate field Q[u],
where u = ξ+ξ^(-1).)
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◆ From: 140.112.246.79
※ 編輯: BugofBook 來自: 140.112.246.79 (06/15 22:24)
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