A workbook in higher algebra by David B Surowski

By David B Surowski

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N}, let F be a field and let V be the F-vector space with ordered basis (v1 , v2 , . . , vn ). As we have already seen, G acts on V via the homomorphism φ : G → GL(V ). Set V G = {v ∈ V | φ(g)v = v}. (a) Show that dim V G = the number of orbits of G on {1, 2, . . , n}. (b) Let V1 ⊆ V be a G-invariant subspace of V ; thus G acts as a group of linear transformations on the quotient space V /V1 . Show that if the field F has characteristic 0 or is prime to the order of |φ(G)|, then (V /V1 )G ∼ = V G /V1G .

Then it is customary to say that G has generators S and relations R, or that G has presentation G = S| r = e, r ∈ R . A simple example is in order here. Let D be a dihedral group of order 2k; thus D is generated by elements n, k ∈ D such that nk = h2 = e, hnh = n−1 . Let F be the free group on the set S = {x, y}. The kernel of the homomorphism F → D determined by x → n, y → h can be shown to be xk , y 2 , (yx)2 (we’ll prove this below). Thus D has presentation D = x, y| xk = y 2 = (xy)2 = e . One need not always write each “relation” in the form r = e.

Define E1 E2 ⊆ E to be the smallest field containing both E1 and E2 . E1 E2 is called the composite (or compositum) of the fields E1 and E2 . Prove that if [E : F] < ∞, then [E1 E2 : F] ≤ [E1 : F] · [E2 : F]. 13. Given a complex number α it can be quite difficult to determine whether α is algebraic or transcendental. It was known already in the nineteenth century that π and √ e are transcendental, but the fact that such numbers as eπ and 2 2 are transcendental is more recent, and follows from the following deep theorem of Gelfond and Schneider: Let α and β be algebraic numbers.

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