[代數] ring, additive abelian group,field
不好意思又來問代數了^^
1. For which of the following rings is it possible for the product
of two nonzero elements to be 0?
(A) The ring of complex numbers
(B) The ring of integers modulo 11
(C) The ring of continuous real-valued functions on [0,1]
(D) The ring {a+b*sqrt(2): a and b are rational numbers}
(E) The ring of polynomials in x with real coefficients.
Ans:C
b/c a field has no 0 divisor, so (A), (B) were deleted.
(D)我不知道是不是field, 感覺不是，但是因為平常在R的運算，
ab=0 則 a=0 or b=0, 所以它沒有zero divisor.
(E)也是和(D)類似的理由
但是我不知道...
為什麼(C)是對的
(D)是不是一個field
2. Up to isomorphism, how many additive abelian groups of order 16
have the property that x+x+x+x=0 for each x in G?
Ans: 3
Q: 不是很懂要怎麼樣才算滿足x+x+x+x=0
Up to isomorphism, abelian groups of order 16有
2,2,2,2
2,2,4,
4,4
2,8
16.
要怎麼判斷哪一個有滿足x+x+x+x=0呢？
3. why "every field contains at least one finite subfield"is not true?
Thanks a lot!!
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