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Not completely sure what the question is.
To compute an inverse in GF(2^8) one can compute one over something.
Do these examples help?
sage: F.<a> = GF(2^8)
sage: b, c = a^8, a^15
sage: b, c
(a^4 + a^3 + a^2 + 1, a^5 + a^2 + a)
sage: d = c / b
sage: d
a^7
sage: aa = 1/a
sage: aa
a^7 + a^3 + a^2 + a
