Abstract Algebra Answers

Use the Fundamental Theorem of Homomorphism for Groups to prove the following theorem, which is called the Zassenhaus (Butterfly) Lemma:

Let H and K be subgroups of a group G and H′ and K′ be normal subgroups of H

and K, respectively. Then

i) H′(H ∩ K′) H′(H ∩ K)

ii) K′(H′∩ K) K′(H ∩ K)

Construct a field with 125 elements.

Check whether or not Q[x] / <8x³ + 6x² - 9x + 24> is a field

Find all the units of Z[ √-7].

Which of the following statements are true, and which are false? Give reasons for your 

answers.

 i) If k is a field, then so is k × k.

 ii) If R is an integral domain and I is an ideal of R, then Char (R) = Char (R / I)

 iii) In a domain, every prime ideal is a maximal ideal. 

 iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors. 

v) If R is a ring and f(x)∈R[x] is of degree n ∈N, then f(x) has exactly n roots in R.

Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that (Q, +) is not cyclic.

a) Find all the units of Z[ − 7]. b) Check whether or not < 8x + 6x − 9x + 24 > [x] 3 2 Q is a field. c) Construct a field with 125 element

Let S= {(x,y)|x,y € R}. How do we show that S is a ring with identity with the operations defined by (x,y) +(u,v) = (x+u, y+v) and (x,y)(u,v) = (xu-yv,xv+yu)?

Which of the following statements are true, and which are false? Give reasons for your answers. i) If k is a field, then so is k × k. ii) If R is an integral domain and I is an ideal of R, then Char (R) = Char (R I). iii) In a domain, every prime ideal is a maximal ideal. iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors. v) If R is a ring and f(x)∈R[x] is of degree n ∈N, then f(x) has exactly n roots in R.

Let S be a set, R a ring and f be a 1-1 mapping of S onto R. Define + and · on S by: )) x y f (f(x)) f(y 1 + = + − )) x y f (f(x) f(y 1 ⋅ = ⋅ − . ∀ x, y∈S Show that ) (S, +, ⋅ is a ring isomorphic to R.