A field is the continuum of values of a quantity as a function of position and time.
The quantity that the field describes may be a scalar or a vector, and the scalar part may be either real- or complex-valued.
In electromagnetics, the electric field intensity \({\bf E}\) is a real-valued vector field that may vary as a function of position and time, and so might be indicated as “\({\bf E}(x,y,z,t)\),” “\({\bf E}({\bf r},t)\),” or simply “\({\bf E}\).” When expressed as a phasor, this quantity is complex-valued but exhibits no time dependence, so we might say instead “\(\widetilde{{\bf E}}({\bf r})\)” or simply “\(\widetilde{{\bf E}}\).”
An example of a scalar field in electromagnetics is the electric potential, \(V\); i.e., \(V({\bf r},t)\).
A wave is a time-varying field that continues to exist in the absence of the source that created it and is therefore able to transport energy.