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The two fields are distinct because K is constructed with an embedding into the complex numbers and H isn't

sage: CC.coerce_map_from(K)

Composite map:
  From: Number Field in a with defining polynomial x^3 + 2*x + 5
  To:   Complex Field with 53 bits of precision
  Defn:   Generic morphism:
          From: Number Field in a with defining polynomial x^3 + 2*x + 5
          To:   Real Lazy Field
          Defn: a -> -1.328268855668609?
        then
          Conversion via _mpfr_ method map:
          From: Real Lazy Field
          To:   Real Field with 53 bits of precision
        then
          Natural map:
          From: Real Field with 53 bits of precision
          To:   Complex Field with 53 bits of precision
sage: CC.coerce_map_from(H) is None
True