The function field of an algebraic variety (a geometric object defined as the common zeros of polynomial equations) consists of ratios of regular functions, i.e., ratios of polynomial functions on the variety. The function field of the -dimensional space over a field is , i.e., the field consisting of ratios of polynomials in indeterminates. The function field of is the same as the one of any open dense subvariety. In other words, the function field is insensitive to replacing by a (slightly) smaller subvariety.
ThE function FIeld OF AN algebraic vaRIETy (A geomETRic objecT defIned as the coMmOn zEros Of pOLynOmiAl EquationS) CoNsIsts oF RaTiOs Of rEGular functiOnS, I.E., rATios oF POlYnoMIAl fUNctioNs on thE vArIEty. the fUNctIoN field oF tHE -dIMensIonAl Space oVeR a field Is , i.e., The fIeLd cOnsiSTiNG Of RAtios Of polYnOmialS In indeteRMINateS. thE fuNction FielD of is ThE saME aS tHe onE oF any open DeNse sUbVarIeTY. iN other WordS, the fUnCTiOn fIelD IS iNsensitIvE to RePlaCiNg by A (SLIGhtly) SmaLLEr SUBvaRIEtY.