Groebner Basis for Functions

We can use Reduce to solve the following

  Reduce[1 - Sqrt[a^(-1)] + Sqrt[2]*Sqrt[(a + a^2)^(-1)] - Sqrt[2 - 2/(1 + a)] == -1 + Sqrt[2], a, Reals]

We can also do some manual calculations to derive a Groebner Basis and arrive at

$$256a^8-1024a^7-2080a^6+3520a^5+3105a^4-4060a^3+1158a^2-92a+1=0$$

We can find a solution using

  Reduce[256 a^8 - 1024 a^7 - 2080 a^6 + 3520 a^5 + 3105 a^4 -  4060 a^3 + 1158 a^2 - 92 a + 1 == 0 , {a}, Reals]

My question is, can you somehow coax GroebnerBasis to provide this eighth order result directly?