Question

Explain me, please. I don't get why they said so.
x1 = 0.85779 . Is it not that x1 = 85779/100000 ? the same with others x2,x3,x4...
why do they say the polynomial doesn't have rational root?

Answer 1

http://www.mathportal.org/calculators/polynomials-solvers/polynomial-roots-calculator.php

Answer 2

@xapproachesinfinity help me, kid

Answer 3

Maybe there is not just 0.85779 that could be just the first 5 decimals
like say pi is 3.14159...

Answer 4

It must be, or it would be rational. That is the root of what polynomial?

Answer 5

:) It is another topic. The original one is "find the minimal polynomial for sin(2pi/5)
I end up with 16t^4-20t^2+5
The last step is proving it is irreducible.

Answer 6

85779/100000 is the same as 0.85779

Answer 7

irreducible in Q. That is it doesn't have a rational root.

Answer 8

I tested by using that tool. It gives me that answer but I don't understand :)

Answer 9

you can use rational roots theorem to show that
16t^4-20t^2+5=0 has no rational roots (by testing them)

Answer 10

just solve it...it is a quadratic in \(t^2\)

Answer 11

What the website is saying is that there rational roots that can be found using the rational root test. It's not saying that 0.58779 is not a rational number

Answer 12

@Miracrown it says: This polynomial has no rational roots that can be found using Rational Root Test.

Answer 13

Yes, so

Answer 14

@Zarkon you mean I let t^2 = x then solve that quadratic to see that there is no rational root?

Answer 15

yes

Answer 16

check the discriminant

Answer 17

I got it. Thank you to all.

Answer 18

Proposition: If \(K\subseteq L\) is a finite field extension, then
a) L is algebraic over K
b) there are finitely many (algebraic) elements a1,a2,......,a_n in L such that L =K(a1,a2,...,a_n)
@Miracrown question follows

Answer 19

If the conclusion of part b is false, show that there exists an infinite sequence a1,a2,....... of elements of L such that \(K\subset K(a_1)\subset K(a_1,a_2)\subset K(a_1,a_2,a_3)\subset........\)
Show that this contradicts the finiteness of \(dim_K(L)\)

Answer 20

My question: How many parts do I have for this problem? 1 or 2?
the first "show" is to show there does not exist the sequence above, and the second "show" is separate to it, right?