[試題] 105上 林惠雯 代數一 第四次小考
課程名稱︰代數(一)
課程性質︰數學系選修 可抵必修代數導論(一)
課程教師︰林惠雯教授
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰105.12.26
考試時限（分鐘）：50 minutes
試題 :
1.
Let A and B be commutative rings with identity such that A is a subset of B. Show that for x in B, if there is a faithful A[x]-module M which is finitely generated as an A-module, then x is integral over A.
2.
Let V be a vector space over s field F with dimV = n. Show that the symmetric algebra generated by V is isomorphic to the polynomial ring F[x_1,...,x_n].
3.
Let ρ: G → GL(V) be a representation of G and W be a G-invariant subspace of V. Show that there exists a G-invariant subspace U of V such that V is isomorphic to the internal direct sum of W and U.
4.
State and show the Schur's lemma.
Or
4.
Give a proof of the theorem on "Orthogonality relations for χ's" or the theorem on "Divisibility."
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正妹也不過就是一組物質波方程式的特解罷了
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