※ 引述《ssss50201 (ssss)》之銘言：
: 繼續請教大家....^^"
: 1. If F is the field of rational numbers, then the number of distinct
: automorphisms of F is 2. An automorphism g of a field F is a 1-1 mapping
: of F onto itself s.t. g(a+b)=g(a)+g(b), and g(ab)=g(a)g(b), for
: all a,b in F.
: 我現在想到的是trivial mapping 和 identity mapping,
trivial mapping有1-1 and onto嗎
: 但是沒有其他的了嗎？怎麼知道只有這兩個呢？
: --------------------------
: 2. If S is a ring that s=s^2 for each s in S. Which of the following
: must be true?
: a. s+s=0 for each s in S
(s+1)=(s+1)^2=s^2+2s+1=s+2s+1 => 2s=0 => s+s=0
: b. (s+t)^2=s^2+t^2 for each s,t in S
(s+t)^2=(s+t), s^2+t^2=s+t => (s+t)^2=s^2+t^2
: c. S is commutative
sstt=(s^2)(t^2)=st=(st)^2=stst => st=ts
Ans:a,b,c都有
: 答案: a,c
: 我的做法如下 但做到一半卡住了
: s=s*s as given,
: add (-s) to both sides, s*s-s=0
: and b/c left and right distribution law hold in a ring
: so s*(s-1)=0,
: 但是到這邊似乎又不能說s=0 or s=1, 因為有可能有0 divisor.
: 非常謝謝幫忙!!
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