The ring \mathbf{B}_{dR} is defined as follows. Let \C_p denote the completion of \overline{\Q_p}. Let :\tilde{\mathbf{E}}^+ = \varprojlim_{x\mapsto x^p} \mathcal{O}_{\C_p}/(p). An element of \tilde{\mathbf{E}}^+ is a sequence (x_1,x_2,\ldots) of elements x_i\in \mathcal{O}_{\C_p}/(p) such that x_{i+1}^p \equiv x_i \!\!\!\pmod p. There is a natural projection map f:\tilde{\mathbf{E}}^+ \to \mathcal{O}_{\C_p}/(p) given by f(x_1,x_2,\dotsc) = x_1. There is also a multiplicative (but not additive) map t:\tilde{\mathbf{E}}^+\to \mathcal{O}_{\C_p} defined by :t(x_,x_2,\dotsc) = \lim_{i\to \infty} \tilde x_i^{p^i}, where the \tilde x_i are arbitrary lifts of the x_i to \mathcal{O}_{\C_p}. The composite of t with the projection \mathcal{O}_{\C_p}\to \mathcal{O}_{\C_p}/(p) is just f. The general theory of
Witt vectors yields a unique ring homomorphism \theta:W(\tilde{\mathbf{E}}^+) \to \mathcal{O}_{\C_p} such that \theta([x]) = t(x) for all x\in \tilde{\mathbf{E}}^+, where [x] denotes the
Teichmüller representative of x. The ring \mathbf{B}_{dR}^+ is defined to be completion of \tilde{\mathbf{B}}^+ = W(\tilde{\mathbf{E}}^+)[1/p] with respect to the ideal \ker( \theta : \tilde{\mathbf{B}}^+ \to \C_p). Finally, the field \mathbf{B}_{dR} is just the field of fractions of \mathbf{B}_{dR}^+. ==Notes==