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Abstract: Let k be any field, be the polynomial ring of n variables over k. For any where each is a homogeneous polynomial of degree i and , define . If I is an ideal in , define to be , the ideal generated by the terminal forms . Using Bezout''s Theorem and Macaulay''s Theorem, we will establish the following. If satisfying that and , then . Actually the above result is equivalent to Bezout''s Theorem, which sheds another perspective of Bezout''s Theorem. These results are valid in also. [Copyright &y& Elsevier]

Let p be a prime number, G a p-group of order ≤ p4, K a field with charK≠p. If the exponent of G is pe and K contains a primitive peth root of unity, then K(V)G is rational (=purely transcendental) over K where ρ:G→GL(V) is any linear representation of G over K.

Let K be any field which may not be algebraically closed, V a finite-dimensional vector space over K, σ∈GL(V) where the order of σ can be finite or infinite.Theorem.If dimKV≤3, then both K(V)〈σ〉 and K(P(V))〈σ〉 are rational (=purely transcendental) over K. Similar results hold for a cyclic affine action.

Let G be any group acting on the Weyl algebra W1(C). We shall propose a method for finding explicit generators of the ring of invariants W1(C)G. This method can be generalized to the case of the ring of invariants of an almost normalizing extension R over a commutative ring C under the linear action of a finite group G, provided that &z.sfnc;G&z.sfnc; is invertible in C and a set of explicit generators of gr(R)G is known.

LetKbe any field and the symmetric groupSnact on the rational function fieldK(x1,…,xn) by σ(xi)≔xσ(i)if σ∈Anand σ(xi)≔xσ(i)−1if σ∉An. We shall show that the fixed fieldK(x1,…,xn)Snis rational overK. Similar results for actions ofSnon the symmetric powers and exterior powers ofV≔⊕ni=1K·xiare valid.

Letkbe any field,s=(aij)1≤i,j≤n+1∈GLn+1(k),K≔k(x1/xn+1,···,xn/xn+1,y1/yn+1,···,yn/yn+1) a field extension of transcendence degree 2noverk. The diagonal action ofsonKis defined as

We shall use the Binet–Minc formula in the theory of permanents to prove David Richman's theorem: LetGbe a finite group acting onA≔R[a1,…,ar], whereRis any commutative ring with 1/|G|!∈R. Then the ring of invariantsAGis generated overRby ∑σ∈Gσ(aα11aα22···aαrr), where α1+···+αr⩽|G|. Applications of permanents to other problems related to invariants are given also.

In this note an elementary proof that every injective morphism from an affine variety into itself is necessarily surjective is given.

Letpbe any prime number,Gbe the cyclic group of orderp2, and Λ≔RGbe the group algebra ofGover a Dedekind domainRsuch thatpRis a maximal ideal inRand bothR[T]/Φp(T) andR[T]/Φp2(T) are Dedekind domains also, where Φn(T) is thenth cyclotomic polynomial. We shall provide a full list of indecomposable Λ lattices constructed in an explicit way. A complete set of invariants of these indecomposable Λ lattices will be determined also, under an additional assumption onR. The above results were obtained by Reiner [Pacific J. Math.78(1978), 467–501] whenR=Z.

LetKbe the finite field withqelements,Vann-dimensional vector space overK,Pthe parabolic subgroup ofGL(V) associated to the flagV≔V0⊃V1⊃…⊃Vr={0},Uthe unipotent radical ofP. Suppose thatGis a subgroup ofGL(V) such that (i)U⊂G⊂P, and (ii) Image {G→GL(V0/V1)×GL(V1/V2)×…×GL(Vr−1/Vr)}=G1×G2×…×Gr, whereGiis a subgroup ofGL(Vi−1/Vi) for 1≤i≤r.

Let $ K$ 0,m,r$ --> $ K = p > 0,m,r$, and $ L$ with degree $ {p^r}m$ $ {\text{Gal(}}L/K) = \left\langle {\sigma ,\tau :{\sigma ^{{p^r}}} = {\tau ^m} = 1,\tau \sigma {\tau ^{ - 1}} = {\sigma ^e}} \right\rangle $. If $ A$-algebra of degree $ {p^r}$, then $ A$, the theorem has been proved by A. A. Albert [1].