Understanding why Z[x]/(2, x^3+1) is isomorphic to F_2[x]/(x^3+1)

Jack12

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Apr 23, 2018
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I think this isomorphism should be obvious, but its not to me because I'm still not that comfortable with quotient rings. I can prove (and mostly understand) that Z[x]/(2) is isomorphic to F_2[x]. But it's not clear to me how to use this to get to Z[x]/(2, x^3+1) is isomorphic to F_2[x]/(x^3+1). I am especially surprised by this isomorphism, because polynomials in Z[x] behave very differently from those in F_2[x]; so I would expect the ideal (x^3+1) in F_2[x] to have nothing to do with the ideal (2, x^3+1) in Z[x].

P.S. I think my issue comes from not thinking about quotient rings in the right way. When I think about them, I always go back to the formal definition of cosets of an ideal. I suspect that there is a more intuitive way to think of them, which would illuminate this problem for me.
 
(Is this a duplicate?) Understanding why Z[x]/(2, x^3+1) is isomorphic to F_2[x]/(x^3

I'm sorry if this is a duplicate. I am new to the site and I wrote this question before; I clicked "Submit New Thread" last time, but I don't see my question in the forums, nor in the profile. Please let me know if this is a duplicate.

I think this isomorphism should be obvious, but its not to me because I'm still not that comfortable with quotient rings. I can prove (and mostly understand) that Z[x]/(2) is isomorphic to F_2[x]. But it's not clear to me how to use this to get to Z[x]/(2, x^3+1) is isomorphic to F_2[x]/(x^3+1). I am especially surprised by this isomorphism, because polynomials in Z[x] behave very differently from those in F_2[x]; so I would expect the ideal (x^3+1) in F_2[x] to have nothing to do with the ideal (2, x^3+1) in Z[x].

P.S. I think my issue comes from not thinking about quotient rings in the right way. When I think about them, I always go back to the formal definition of cosets of an ideal. I suspect that there is a more intuitive way to think of them, which would illuminate this problem for me.
 
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