Question

If ${\bf {\sigma}_1}$ , ${\bf {\sigma}_2}$ , ${\bf \cdots \cdots}$ , ${\bf {\sigma}_n}$ are distinct automorphism of ${\mathbb K}$ , then show that it is impossible to find elements ${\bf a_1}$ , ${\bf a_2}$ , ${\bf \cdots \cdots}$ , ${\bf a_n}$ not all zero in ${\mathbb K}$ such that ;

${\boxed{\bf a_{1} \sigma_{1} ( u ) + a_{2} \sigma_{2} + \cdots \cdots + a_{n} \sigma_{n} ( u ) = 0 \quad \forall \bf u \in \mathbb K}}$

Topic :- Advance Abstract Algebra MSc 1st sem.​

Answer 1

Answer:

If ${\bf {\sigma}_1}$ , ${\bf {\sigma}_2}$ , ${\bf \cdots \cdots}$ , ${\bf {\sigma}_n}$ are distinct automorphism of ${\mathbb K}$ , then show that it is impossible to find elements ${\bf a_1}$ , ${\bf a_2}$ , ${\bf \cdots \cdots}$ , ${\bf a_n}$ not all zero in ${\mathbb K}$ such that ;

${\boxed{\bf a_{1} \sigma_{1} ( u ) + a_{2} \sigma_{2} + \cdots \cdots + a_{n} \sigma_{n} ( u ) = 0 \quad \forall \bf u \in \mathbb K}}$

Topic :- Advance Abstract Algebra MSc 1st sem.

Step-by-step explanation:

hope helpful

Answer 2

Answer:

A borrowed 3000

interest

=

r

B borrowed 3500

n= 3

Interest paid by

B=

100

pnr

=

100

(3500)(3)(r)

= 105r

Interest paid by

A=

100

pnr

=

100

(3000)(3)(r)

= 90r

Given,

105r−

90r=

150

r(15)= 150

⇒ r= 10

∴ rate of interest

=

10%

per annum.