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Dear Mathematica Gurus!

Who know how to execute SymbolicPolynomialMod
PolynomialMod working only on numbers (not for symbols)

If we know that a,b,c,d,f are five roots of general quintic polynomial 
X^5+t X^4+p X^3+q X^2+r X+s==0

Then we can do SymbolicPolynomialMod of g (manually not by Mathemathica 

g=(a^6 b^7 c^5 d^2 + a^7 b^5 c^6 d^2 + a^5 b^6 c^7 d^2 +
 a^7 b^6 c^2 d^5 + a^2 b^7 c^6 d^5 + a^6 b^2 c^7 d^5 +
 a^5 b^7 c^2 d^6 + a^7 b^2 c^5 d^6 + a^2 b^5 c^7 d^6 +
 a^6 b^5 c^2 d^7 + a^2 b^6 c^5 d^7 + a^5 b^2 c^6 d^7 -
 a^6 b^8 c^3 d^2 f - a^8 b^3 c^6 d^2 f - a^3 b^6 c^8 d^2 f -
 a^8 b^6 c^2 d^3 f - a^2 b^8 c^6 d^3 f - a^6 b^2 c^8 d^3 f -
 a^3 b^8 c^2 d^6 f - a^8 b^2 c^3 d^6 f - a^2 b^3 c^8 d^6 f -
 a^6 b^3 c^2 d^8 f - a^2 b^6 c^3 d^8 f - a^3 b^2 c^6 d^8 f +
 a^7 b^6 c^5 f^2 + a^5 b^7 c^6 f^2 + a^6 b^5 c^7 f^2 -
 a^8 b^6 c^3 d f^2 - a^3 b^8 c^6 d f^2 - a^6 b^3 c^8 d f^2 -
 a^6 b^8 c d^3 f^2 - a^8 b c^6 d^3 f^2 - a b^6 c^8 d^3 f^2 +
 a^6 b^7 d^5 f^2 + a^7 c^6 d^5 f^2 + b^6 c^7 d^5 f^2 +
 a^7 b^5 d^6 f^2 - a^8 b^3 c d^6 f^2 - a b^8 c^3 d^6 f^2 +
 b^7 c^5 d^6 f^2 + a^5 c^7 d^6 f^2 - a^3 b c^8 d^6 f^2 +
 a^5 b^6 d^7 f^2 + a^6 c^5 d^7 f^2 + b^5 c^6 d^7 f^2 -
 a^3 b^6 c d^8 f^2 - a^6 b c^3 d^8 f^2 - a b^3 c^6 d^8 f^2 -
 a^6 b^8 c^2 d f^3 - a^8 b^2 c^6 d f^3 - a^2 b^6 c^8 d f^3 -
 a^8 b^6 c d^2 f^3 - a b^8 c^6 d^2 f^3 - a^6 b c^8 d^2 f^3 -
 a^2 b^8 c d^6 f^3 - a^8 b c^2 d^6 f^3 - a b^2 c^8 d^6 f^3 -
 a^6 b^2 c d^8 f^3 - a b^6 c^2 d^8 f^3 - a^2 b c^6 d^8 f^3 +
 a^6 b^7 c^2 f^5 + a^7 b^2 c^6 f^5 + a^2 b^6 c^7 f^5 +
 a^7 b^6 d^2 f^5 + b^7 c^6 d^2 f^5 + a^6 c^7 d^2 f^5 +
 a^2 b^7 d^6 f^5 + a^7 c^2 d^6 f^5 + b^2 c^7 d^6 f^5 +
 a^6 b^2 d^7 f^5 + b^6 c^2 d^7 f^5 + a^2 c^6 d^7 f^5 +
 a^7 b^5 c^2 f^6 + a^2 b^7 c^5 f^6 + a^5 b^2 c^7 f^6 -
 a^8 b^3 c^2 d f^6 - a^2 b^8 c^3 d f^6 - a^3 b^2 c^8 d f^6 +
 a^5 b^7 d^2 f^6 - a^3 b^8 c d^2 f^6 - a^8 b c^3 d^2 f^6 +
 a^7 c^5 d^2 f^6 + b^5 c^7 d^2 f^6 - a b^3 c^8 d^2 f^6 -
 a^8 b^2 c d^3 f^6 - a b^8 c^2 d^3 f^6 - a^2 b c^8 d^3 f^6 +
 a^7 b^2 d^5 f^6 + b^7 c^2 d^5 f^6 + a^2 c^7 d^5 f^6 +
 a^2 b^5 d^7 f^6 + a^5 c^2 d^7 f^6 + b^2 c^5 d^7 f^6 -
 a^2 b^3 c d^8 f^6 - a^3 b c^2 d^8 f^6 - a b^2 c^3 d^8 f^6 +
 a^5 b^6 c^2 f^7 + a^6 b^2 c^5 f^7 + a^2 b^5 c^6 f^7 +
 a^6 b^5 d^2 f^7 + b^6 c^5 d^2 f^7 + a^5 c^6 d^2 f^7 +
 a^2 b^6 d^5 f^7 + a^6 c^2 d^5 f^7 + b^2 c^6 d^5 f^7 +
 a^5 b^2 d^6 f^7 + b^5 c^2 d^6 f^7 + a^2 c^5 d^6 f^7 -
 a^3 b^6 c^2 d f^8 - a^6 b^2 c^3 d f^8 - a^2 b^3 c^6 d f^8 -
 a^6 b^3 c d^2 f^8 - a b^6 c^3 d^2 f^8 - a^3 b c^6 d^2 f^8 -
 a^2 b^6 c d^3 f^8 - a^6 b c^2 d^3 f^8 - a b^2 c^6 d^3 f^8 -
 a^3 b^2 c d^6 f^8 - a b^3 c^2 d^6 f^8 - a^2 b c^3 d^6 f^8)

by substitutions g/. {a^5->-t a^4-p a^3-q a^2-r a-s,b^5->-t b^4-p b^3-q 
b^2-r b-s,
c^5->-t c^4-p c^3-q c^2-r c-s,d^5->-t d^4-p d^3-q d^2-r d-s,f^5->-t 
f^4-p f^3-q f^2-r f-s}


How we can do full SymbolicPolynomialMod on the form g after such 
procedure will be not higher degree as 4 for a,b,c,d,f ?

Best wishes

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