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抽代补考 1 (10)show that if f : XY is a bijection , then it has exactly one inverse . 2 (10)Define f:{0,1,2,…,10} by f(n)=the remainder dividing 4n2-3n7 by 11 . (i) Show that f is a permutation . (ii) Compute the parity of f . (iii) Compute the inverse of f . (20)Let G be a finite group with KG . If (,[G:K])=1,prove that K is the unique subgroup of G having order . (20)Let R be a PID .Show that every α, β has a gcd δ,which is a linear combination ofα andβ : δ=aα+bβ ,where a,b R . (10)Let f(x) E[x] ,where E is field , and let :EE be an automorphism .If f(x) splits and fixes every root of f(x) ，prove that fixes every coefficient of f(x) . (20)If R is commutative notherian ring , then R[x] is also notherian . 7 (10) Given a pushout diagram in R Mod A C f β B D Prove that g injective impliesα injective . 抽代期末 1. If a1，a2 ，… , at-1 , at are elements in group G , prove that (a1a2…at-1at)-1 = a -1 ta-1 t-1…a -1 2 a-1 1.(10) 2. Let G be a finite group , H be its subgroup . Prove that is a divisor of .(10) 3. Prove that every educlidean ring R is a PID .(20) 4. Let k be a field .Show that every f(x),g(x) k[x] has a gcd h(x) ,which is a linear combination of f(x) and g(x) :h(x)=a(x)f(x)+b(x)g(x) where a(x) ,b(x) k[x] .(20) 5. Let f(x)=x5 -4x+2Q(x) and let G be its Galois group .(20) i Assming that f(x) is an irreducible polynomial , prove that is a multiple of 5 . ii Prove that f(x) has three real roots and two complex roots , which are , of course , Complex Conjugates . Conclude that if the Galois group G of f(x) is viewed as a subgroup of S5 , then G Contains complex conjugation , which is a transposition of the roots of f(x). iii Prove that GS5 , and conclude that f(x) is not solvable by radicals . 6. An ideal I in a commutative ring R is a prime ideal if and only if R|I is a domain .(10) 7. Given a pushout diagram in R Mod A C