How to sum or product of two elements of two different fields?

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I would like to sum of product two elements from different fields and the resultant operation should be in the bigger field. I have used the following codes, but it gives errors.

R.<x> = PolynomialRing(GF(2))
F = GF(2^3, name='x', modulus=x^3 + x^2 + 1)
G = GF(2^8, name='x', modulus=x^8 + x^5 + x^3 + x + 1)
print F.fetch_int(3)
print  G.fetch_int(5)
#F.fetch_int(3) => x + 1
#G.fetch_int(5) => x^2 + 1
#print F.fetch_int(3)*G.fetch_int(5) 
print (x^2 + 1)*(x + 1)