[代數] automorphism, ring
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,
但是沒有其他的了嗎？怎麼知道只有這兩個呢？
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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
b. (s+t)^2=s^2+t^2 for each s,t in S
c. S is commutative
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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