Download e-book for kindle: Abstract Algebra (Holden-Day Series in Mathematics) by Andrew O Lindstrum

By Andrew O Lindstrum

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Additional resources for Abstract Algebra (Holden-Day Series in Mathematics)

Example text

Gr ] is a finitely generated F-algebra. In the notation of the above proof, the elements f1 , f2 , . . , fn are traditionally called primary invariants and the elements g1 , g2 , . . , gr secondary invariants. It is important to notice that there are infinitely many choices for both the primary and secondary invariants. These two terms merely describe the role of certain invariants within a particular fixed generating set. Since F[V ]G is finitely generated, its spectrum, Spec(F[V ]G ) is an affine variety which we denote by V //G .

Ft may be taken in R1 then we say that R is a standard graded K-algebra. , every ideal I of R is finitely generated: I = (g1 , g2 , . . , gs ) = Rg1 + Rg2 + · · · + Rgs . An element h ∈ R is homogeneous (of degree d) if there exists d such that h ∈ Rd . An ideal I of R is graded (or homogeneous) if it generated by homogeneous elements. Equivalently, I is graded if whenever h ∈ I and we write h = h0 + h1 + · · · + hd with hi ∈ Ri , we have all h1 , h2 , . . , hd ∈ I. The ideal R+ := ⊕∞ d=1 Rd is the unique maximal homogeneous ideal in R.

Gm = G with Gi /Gi−1 ∼ = Cp for all i = 1, 2, . . , m. A consequence of this lemma is that we may compute the invariants of a p-group G by repeatedly computing invariants under an action of the cyclic group Cp . Given {e} = G0 G1 G2 . . Gm = G, we proceed as follows. First, we compute R1 := K[V ]G1 where G1 ∼ = Cp . Then we compute R2 = C K[V ]G2 = (K[V ]G1 )G2 /G1 ∼ = R1 p . Continuing in this manner we compute C Rj+1 = K[V ]Gj+1 = (K[V ]Gj )Gj+1 /Gj ∼ = Rj p for j = 0, 1, . . , m. This yields Rm+1 = K[V ]G .

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Abstract Algebra (Holden-Day Series in Mathematics) by Andrew O Lindstrum


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