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Indeed, in Sage the name of the undeterminates matters. You can define the mapping as follows:

sage: h = bRing.hom([aa])
sage: h
Ring morphism:
  From: Number Field in a with defining polynomial b^2 + 1
  To:   Number Field in a with defining polynomial a^2 + 1
  Defn: a |--> a
sage: h(bb) == aa
True

Indeed, in Sage the name of the undeterminates matters. matters, this is not a bug. Think about the multivariate case to be convinced of the reason why.

You can define the mapping as follows:

sage: h = bRing.hom([aa])
sage: h
Ring morphism:
  From: Number Field in a with defining polynomial b^2 + 1
  To:   Number Field in a with defining polynomial a^2 + 1
  Defn: a |--> a
sage: h(bb) == aa
True

Indeed, in Sage the name of the undeterminates indeterminates matters, this is not a bug. Think about the multivariate case to be convinced of the reason why.

You can define the mapping as follows:

sage: h = bRing.hom([aa])
sage: h
Ring morphism:
  From: Number Field in a with defining polynomial b^2 + 1
  To:   Number Field in a with defining polynomial a^2 + 1
  Defn: a |--> a
sage: h(bb) == aa
True