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We present results for the fermion determinant in an external Yukawa-coupledscalar field, a quantity relevant to instanton and sphaleron transition ratesas well as bubble nucleation occuring in the electroweak phase transition whenthe scalar field is identified with the Higgs field. We calculate thedeterminant exactly and find an approximation (essentialy of the gradientexpansion type) that reproduces the exact results very well if the fermion mass$m_F$ becomes larger than the inverse of the size $R$ of the background fieldconfiguration. The results are given for 2, 3 and 4 Euclidean dimensions.Comment: Dortmund University DO-TH-94/04, 16 pages, LaTeX, macros: epsfig.sty  7 figures appended as uuencoded, tar-compressed .eps file

The fermion determinant in external scalar fields: exact results and large mass approximation