Abstract Algebra: Theory and Applications · Problem 17.5.16

Integral Domains — Thomas Judson

Mathematics — Abstract Algebra · polynomial-ring-isomorphism

Problem

Suppose that RR and SS are isomorphic rings. Prove that R[x]≅S[x]R[x] \cong S[x]. Hint. Let ϕ:R→S\phi: R \to S be an isomorphism. Define ϕˉ:R[x]→S[x]\bar{\phi}: R[x] \to S[x] by ϕˉ(a0+a1x+⋯+anxn)=ϕ(a0)+ϕ(a1)x+⋯+ϕ(an)xn\bar{\phi}(a_0 + a_1 x + \cdots + a_n x^n) = \phi(a_0) + \phi(a_1)x + \cdots + \phi(a_n)x^n.

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