This is a closed-book exam. Read each question carefully, and show all your work

Question

This is a closed-book exam. Read each question carefully, and show all your work. Every group of prime order is commemorative. S_t contains an element of order 8 The groups Z_2 times Z_2 times Z_2, and Z_a times z_12 is 15. are isomorphic. The maximal order of an element of z_a times z_15 is 15. Construct an explicit injective homomorphism Z_a rightarrow the image of each element of Z_a The or false. A maximal ideal is a prime ideal If F is an of complex numbers c then F = C Every field is contained in a strictly larger field. [Q(3^1/3 + 3^2/3) Q| = 25. Show that alpha = squareroot 1 + squareroot 3 is algebraic over Q by finding the monic irreducible polynomial for a over Q. Prove that the polynomial you found is indeed irreducible. Determine [Q(alpha); Q| Explain bow to construct a field with 9 elements. Give an explicit description of this field Prove that I is a prime ideal of the commutative ring R if and only if R/I is an integral domain

Explanation / Answer

a)- true, as every group of prime order is abelian group and abelian group is commutitive

b)- false, because S7 doesnot have any pairs whose L.C.M will be 8

c)-true ,order of both group are same

d)- false because the maximal order element in Z2 ×Z15 is 30

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