ALGEBRA Sample Clauses

ALGEBRA. ‌ In this annex the pedagogical contents and challenges of the Algebra Lesson Plan are fully described. This is a basic example of the contents that can be “slotted” into Game Plots, and relies mostly on minigames and quizzes to provide evidence and progress.

ALGEBRA. Knowledge of the central issues of algebra up to and including the equations of the second degree with one unknown variable. Geometry: Knowledge of the central issues of two and three-dimensional geometry. (▇▇▇▇▇▇▇, 1979, p. 19)

ALGEBRA. For any abelian group M we set Mp := Zp ⊗Z M and write Mp∧ for its pro-p completion lim M/pnM (so that Mp = Mp∧ if M is finitely generated). For any natural number n we write M [n] for the subgroup of M comprising those elements that have order dividing n. We write Mtor for the union of M [n] over all natural numbers n, set M := M/Mtor and identify M with a sublattice of the vector space Q ⊗Z M in the obvious way. If M is a finitely generated Zp-module we define its ‘Zp-rank’ to be rkZp (M ) := dimQp (Qp ⊗Zp M ) and its ‘p-rank’ to be rkp(M ) := dimFp (M/pM ). We note that rkp(M ) = dimFp (M [p]) +rkZp (M ) and that for any exact sequence of finitely generated Zp-modules M1 −θ→1 M2 −θ→2 M3 one has rkp(Im(θi)) ≤ rkp(M2) ≤ rkp(M1) + rkp(M3) for both i = 1 and i = 2. We often abbreviate the vector space Qp ⊗Zp M to Qp ·M . If Γ is a finite group, then a finitely generated Zp[Γ]-module M is said to be a ‘Zp[Γ]- lattice’ if Mtor vanishes. Each Zp[Γ]-lattice M can therefore be regarded as a submodule of the space Qp ·M .

ALGEBRA. ‌ For a unital ring A, we write A× for the multiplicative group of invertible elements of A and ζ(A) for the centre of A. By an A-module, we shall always mean a left A-module. For the inclusion of rings A ⊂ A′, and an A-module M , we write MA' := A′ ⊗A M . For each pair of A-modules M and M′ we write IsA(M, M′) for the set of A-module isomorphisms from M to M′ and AutA(M ) for the group of A-module automorphisms of M . Fix a finite group Γ, and a Dedekind domain R of characteristic zero, with field of fractions F , and E is an extension field of F . We let OE denote the ring of integers of a field E. For any R[Γ]-module M , we write ME := E⊗RM , and Mv := Rv ⊗RM , with Rv the completion of R at v, where v is a (non-zero) prime ideal of R. Let E/F be a finite Galois extension of fields. We let Gal(E/F ) denote the Galois group of E/F , and we write the action of Gal(E/F ) on E by x '→ g(x) for x ∈ E and g ∈ Gal(E/F ). We fix a separable closure Fc of F , and write ΩF for the absolute Galois group Gal(Fc/F ) of

ALGEBRA. (One of the following) 1. Algebra II in high school with at least a C- grade. 2. MATH 096 Intermediate Algebra MATH 151 Elementary Algebra