An explicit formula for the Hilbert symbol of a formal group
  Abrashkin established the Bruckner-Vostokov formula for the Hilbert symbol of a formal group under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of ($\varphi, \Gamma$)-modules and a cohomological interpretation of Abrashkin's technique. To do this, we build ($\varphi, \Gamma$)-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas for the cup-product and the Kummer map. 
Floric Tavares Ribeiro