[代數] 習題幾問

看板Math作者jacky7987 (憶)時間15年前 (2011/01/03 23:55)推噓1(1推 0噓 6→)

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大致上是ring的問題 1. Let R be a (integral) domain and F=Frac(R),the fraction(quotient) field of R.Prove that Frac(R[x]) is isomorphic to F(x)=Frac(F[x]) 原本我就知道 f:R ----> F r |---> [r,1] 是個homo 所以就想說 g:Frac(R[x])-> F(x) ~ ~ ~ f(x)/g(x) |-> f(x)/g(x) f(x)是把f(x)的係數都變成 [a_i,1] 證明他是isom.這樣對嗎?還是有別的作法? 2.K is a field and K[[x]] denote the set of all power series.Let f in K[[x]] inf f=sum(a_i)x^i and ord(f)=m i=0 ,where m is the smallest natural number for which a_m is not zero. If f is a unit iff ord(f)=0 i.e. the constant is not 0. =>這個方向的我會證明了(用到ord(fg)=ord(f)+ord(g)) <=我沒有甚麼頭緒似乎是要證明1/f可以寫成power series不過我不知道要怎麼證明 3. If p is a prime and m,n in |N,prove pm m ( )≡( ) mod p pn n 先謝謝大家ˊˋ -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 123.193.93.138

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