If F is a field, show that U(F[x]) = F*.
Let $F$ be a field. First observe that every nonzero constant polynomial $a\in F^\times$ is a unit in $F[x]$: its inverse is _____ _________ _______ ______ ___ __________ _______ ____ ___ ____ ____.
________ _________ ____ ______ ____ __________ ___ ____ ________.
___ _____ _________ ________ _______ ____ __________.
__________ ______ _____ ______ ______.
______ ____ ___ _______ _______ _______.
___ ____ ____ ____ ________ _____ _______.
__________ ______ ___ __________ __________ _____ ___ ______ ______ ___.
_______ __________ ______ _____ _____ _____ ______ ___ _______ ________ ________ ______.
___ __________ _____ _____ _____ _______ __________ ____ _______ ______.
_______ ___ ________ ________ ____ ________ ____ ________ ____ _____.
___ _____ _________ _________ ________ _____ _______ _________.
______ __________ _________ _____ _____ ______ ________ _______.
______ _________ _______ __________ __________ ___ _______ _________.
Get Full Answer on WhatsApp
Which of the following statements are true and which are false? Justify your answer with a short proof or a counter example.
(a) Every subgroup of S3 is normal.
(b) Every abelian group is cyclic.
(c) In a ring with unity, the sum of any two units is a unit.
(d) If a field has characteristic p, p a prime, the field is finite.
(e) If every element in group has finite order, the group is finite.
Define a subgroup of a group. Check whether
is a subgroup of the the group of 2 × 3 matrices over ℂ under addition
Let R = Z20
(i) Give, with justificaiton, a nilpotent element in R.
(ii) Give, with justification, a zero divisor in R which is not nilpotent.
(iii) What is the order of U(R)?
Let S¹ = {z ∈ C* | |z| = 1} and Un = {z ∈ C* | z" = 1} for n ∈ N. Check that Un ⊆ S¹. Further, show that Un ≤ S¹.
Show that is not a principal ideal in Z[x].
Let A be a 3 x 4 real matrix, B be a 4 × 2 real matrix and C be a 2 x 3 real matrix. Which of the following operations are defined?
(i) CA + B¹
(ii) AB + C¹
For those operations that are defined, what is the order of the resulting matrix?
If H and K are normal abelian subgroups of a group, and if H∩K = {e}, show that HK is abelian. Will the result be still true if we remove the condition that H and K are normal? Justify your answer.