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Kernal and image homorphism q

Stuck on part c and d. What justification would it be for c. Its so obvious.

Also please can someone check if I have done part b correctly.

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Original post by cooldudeman
Stuck on part c and d. What justification would it be for c. Its so obvious.

Also please can someone check if I have done part b correctly.

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For c, you should provide a polynomial whose image is the particular diagonal matrix {{a,b,c},{0,a,b},{0,0,a}}.

For b, I don't like your first direction, but the second direction is fine. You seem to be assuming a lot of what you need to prove. I think it can be fixed by reordering your sentences, and changing an "if" to an "iff". You need to show that a_0, a_1, a_2 are 0, not assume it (sentence starting "Since").
You've also assumed that p has degree 3 or more, and you're reusing the symbol a_0 to mean both the x^0 coefficient and the x^3 coefficient.

d follows from the first isomorphism theorem.
Original post by Smaug123
For c, you should provide a polynomial whose image is the particular diagonal matrix {{a,b,c},{0,a,b},{0,0,a}}.

For b, I don't like your first direction, but the second direction is fine. You seem to be assuming a lot of what you need to prove. I think it can be fixed by reordering your sentences, and changing an "if" to an "iff". You need to show that a_0, a_1, a_2 are 0, not assume it (sentence starting "Since").
You've also assumed that p has degree 3 or more, and you're reusing the symbol a_0 to mean both the x^0 coefficient and the x^3 coefficient.

d follows from the first isomorphism theorem.


For b, is it now OK? Also how do we know n-3 is >=0?
Looking at c and d now..
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(edited 9 years ago)
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This is what I have for part c. I cant think anything else but to use the same idea.

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Original post by cooldudeman
For b, is it now OK? Also how do we know n-3 is >=0?
Looking at c and d now..
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Your b answer is now good, I think. n-3 >= 0, because if it weren't then n would be 0 or 1 or 2, and in none of those cases can theta be 0 (unless the polynomial itself were 0).
Original post by cooldudeman
This is what I have for part c. I cant think anything else but to use the same idea.

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I'd say that's fine. My answer would be the single line "p=aX2+bX+cp = a X^2+b X+c has θ(p)\theta(p) being {{a,b,c},{0,a,b},{0,0,a}}", but you've also proved containment the other way, which is strictly more correct than mine.
Original post by Smaug123
I'd say that's fine. My answer would be the single line "p=aX2+bX+cp = a X^2+b X+c has θ(p)\theta(p) being {{a,b,c},{0,a,b},{0,0,a}}", but you've also proved containment the other way, which is strictly more correct than mine.


thanks. can you help me on part d more please... I know theta has gotta be a bijection to be an isomorphism, so R[X]/(X^3) has gotta be a bijection but this is not really a function. its a quotient ring.

Edit: Just read the correct theorem. Done it.
(edited 9 years ago)
Original post by cooldudeman
thanks. can you help me on part d more please... I know theta has gotta be a bijection to be an isomorphism, so R[X]/(X^3) has gotta be a bijection but this is not really a function. its a quotient ring.

Edit: Just read the correct theorem. Done it.

It's a really key thing, the first isomorphism theorem: the idea of images of homomorphisms as quotients, with the kernel being the corresponding normal subgroup, is an idea which comes up everywhere.

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