作者ssss50201 (ssss)
看板Math
標題[代數] ring, additive abelian group
時間Fri Nov 9 13:02:31 2012
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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◆ From: 108.3.154.49
→ agga :第二題, for each x 所以不可以有元素的order超過4 11/09 13:42
推 turboho :1C考慮一個函數的support在[0,0.5]另一個在[0.5,1] 11/09 13:46
→ turboho :3考慮field with char = 0 11/09 13:47
→ turboho : 如果要包涵某個non-0 elt就要包含無窮多個 11/09 13:48
→ turboho : i.e. x, 2x, 3x, ... 11/09 13:48
→ turboho :1DE recall:subring of domain/field is domain 11/09 13:49
→ ssss50201 :謝謝～ 這樣1&2都瞭解了!! yeah!! 11/09 14:29
→ ssss50201 :但是3還是有點不懂。。。 11/09 14:29
→ ssss50201 :field with char = 0表示unity不管加幾次都不是零 11/09 14:31
→ ssss50201 :阿 是任何一個元素不管自己加自己幾次都不會是0 11/09 14:32
→ ssss50201 :為什麼說"如果要包涵某個non-0 elt就要包含無窮多個" 11/09 14:32
→ ssss50201 :為什麼特別考慮了" 包涵某個non-0 element" ? 11/09 14:33
推 jacky7987 :如果它包含有限子體的話 他特徵數就不是0 11/09 21:31