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Let R be a ring with 1, ρ an automorphism of R of order 2. Then a normal extension of the free quadratic extension R[x,ρ] with a basis {1,x} over R with an R-automorphism group G is characterized in terms of the element (x−(x)α) for α in G. It is also shown by a different method from the one given by Nagahara that the order of G of a Galois extension R[x,ρ] over R with Galois group G is a unit in R. When 2 is not a zero divisor, more properties of R[x,ρ] are derived

On automorphism group of free quadratic extensions over a ring