MATH 113 Study Guide - Final Guide: Commutator Subgroup, Commutative Ring, Abelian Group
MATH 113: SAMPLE FINAL
1. (a) By using Euler’s theorem, compute 275( mod 21). You need to state the
theorem clearly. (b) Determine the number of solutions for 15x=3( mod 21) in
Z21.
2. Let σ, τ denote the permutations of S7given by (1,2,3,4)(5,6,7) and (1,7)(2,4,5,3,6).
(a)Compute στ and τσ. (b)Does there exist γ∈S7with γσγ−1=τ? Why?
3. True or False questions:
(a) There exists no abelian subgroup in S7with order 15.
(b)In a commutative ring, the intersection of two prime ideals is still a prime
ideal.
(c)An inner automorphism of an abelian group must be the identity map.
(d)The commutator subgroup of a simple group Gmust be Gitself.
(e)x14 =1 has exactly 2 solutions in the field Z113.
(f)115x8+6x5+2016x2+6 is irreducible over Q
(g)If Ris a ring and f(x) and g(x) are in R[x] with degree 2 and 3, then f(x)g(x)
must have degree 5.
4. (a)Let Rbe a commutative ring with 1 ,0. Prove by definition that Ris an
integral domain if and only if {0}is a prime ideal of R. (b) Let Rbe a commutative
ring and Iis an ideal. Define √I={x∈R|xn∈Ifor some n∈Z>0}. Prove √Iis
an ideal. (c) Prove if Iis a prime ideal in a commutative ring R, then √I=I.
5. Let α=q2+√2∈R. (a) Find the irreducible polynomial irr(α, Q) over Q.
(b)Find a basis of Q(α) over Q. (c)Prove x3=5 has no solution in Q(α).
6. Determine whether x4+x+1 is irreducible over Q. Explain.
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Document Summary
Math 113: sample final: (a) by using euler"s theorem, compute 275( mod 21). You need to state the theorem clearly. (b) determine the number of solutions for 15x = 3( mod 21) in. Prove by de nition that r is an integral domain if and only if {0} is a prime ideal of r. (b) let r be a commutative. I is ring and i is an ideal. De ne an ideal. (c) prove if i is a prime ideal in a commutative ring r, then. I = {x r|xn i for some n z>0}. 2 r. (a) find the irreducible polynomial irr( , q) over q. (cid:113: let = 2 + (b)find a basis of q( ) over q. (c)prove x3 = 5 has no solution in q( ): determine whether x4 + x + 1 is irreducible over q.
