Show that the rings Z[x] and Z are not isomorphic

  • #1
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Homework Statement


Show that the rings Z[x] and Z are not isomorphic

Homework Equations




The Attempt at a Solution


I want to show that these are not isomorphic. The thing is that I already know that ##\mathbb{Z}/(x) \cong \mathbb{Z}##, but for some reason I can't find specific structural properties of ##\mathbb{Z}[x]## that are different than ##\mathbb{Z}##
 

Answers and Replies

  • #3
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##\mathbb{Z}/(x) \cong \mathbb{Z}##?
I meant to write ##\mathbb{Z}[x]/(x) \cong \mathbb{Z}##
 
  • #4
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What if you assumed an isomorphsm ##\varphi\, : \,\mathbb{Z}[x] \longrightarrow \mathbb{Z}##. Then ##\varphi((x)) ## is an ideal in ##\mathbb{Z}##, say ##\varphi((x)) = n\mathbb{Z}## with ##\varphi(x)=n##. What do you get if you factor this ideal on both sides?
 
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  • #5
15,129
12,838

Homework Statement


Show that the rings Z[x] and Z are not isomorphic

Homework Equations




The Attempt at a Solution


I want to show that these are not isomorphic. The thing is that I already know that ##\mathbb{Z}/(x) \cong \mathbb{Z}##, but for some reason I can't find specific structural properties of ##\mathbb{Z}[x]## that are different than ##\mathbb{Z}##
Another idea is to use the fact that ##\mathbb{Z}[x]## is no PID, e.g. ##I:=\langle 2, 1+x \rangle##.
 
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