Let Q be a nondegenerate quadratic form over the nite eld Fq of odd prime power order
q with charFq 6= 2 and Let On(Fq) be the associated orthogonal group. Let On(Fq) act
linearly on the polynomial ring Fq[x1; : : : ; xn]: In this thesis we nd the invariant subring
Fq[x1; x2; x3; x4; x5]O5(Fq ) with explicit generators. We also prove that this subring is a UFD.

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