Construct a field with 125 elements.
A field with 125 elements must have order (125 = 5^3), so it is a finite field of characteristic 5. Denote it by (\mathbb{F}_{5^3}). Start with the prime field (\mathbb{F}*5 = {0,1,2,3,4}). To ____ _________ ___ _______ _________ _______ ____ __________ _________ ____ __________ _________.
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Is , for any two ideals I and J of a ring R? Give reasons for your answer. Let S be a set, R a ring and f be a 1-1 mapping of S onto R. Define + and
on S by:
.
Show that is a ring isomorphic to R.
Find all the units of .
Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that is not cyclic.
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)
ii)
iii)
The situation can be represented by the subgroup diagram below, which explains the name ‘butterfly’.
PART-B (MM: 30 Marks)
(Based on Block 3.)
Using Cayley’s theorem, find the permutation group to which a cyclic group of order 12 is isomorphic.
Is , for any two ideals I and J of a ring R? Give reasons for your answer.
Construct a field with 125 elements.