# Abstract Algebra- Finding the Minimal Polynomial

## Homework Statement

Given field extension C of Q, Find the minimal polynomial of a=sqrt( 5 + sqrt(23) ) (element of C).

## The Attempt at a Solution

I may be complicating things, but let me know if you see something missing.

Doing the appropriate algebra, I manipulated the above expression into (a^2 - 5)^2=23

Expanding the left side, we get a^4 - 10*a^2 + 25 = 23 , i.e. a^4 - 10*a^2 + 2 = 0

So I plan to use f(x)=x^4 - 10*x^2 + 2

From here, I just need to show that it's irreducible.

If it is reducible, there will be either a linear factor or a quadratic factor.

My last step was simply to just use brute force to find a contradiction when comparing the following expressions with my polynomial above:

(ax+b)(cx^3 + dx^2 + ex + f) and

(ax^2+bx+c)(dx^2+ex+f)

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

Given field extension C of Q, Find the minimal polynomial of a=sqrt( 5 + sqrt(23) ) (element of C).

## The Attempt at a Solution

I may be complicating things, but let me know if you see something missing.

Doing the appropriate algebra, I manipulated the above expression into (a^2 - 5)^2=23

Expanding the left side, we get a^4 - 10*a^2 + 25 = 23 , i.e. a^4 - 10*a^2 + 2 = 0

So I plan to use f(x)=x^4 - 10*x^2 + 2

From here, I just need to show that it's irreducible.

If it is reducible, there will be either a linear factor or a quadratic factor.

My last step was simply to just use brute force to find a contradiction when comparing the following expressions with my polynomial above:

(ax+b)(cx^3 + dx^2 + ex + f) and

(ax^2+bx+c)(dx^2+ex+f)

Isn't f(x) a quadratic in ##y = x^2?##

STEMucator
Homework Helper

## Homework Statement

Given field extension C of Q, Find the minimal polynomial of a=sqrt( 5 + sqrt(23) ) (element of C).

## The Attempt at a Solution

I may be complicating things, but let me know if you see something missing.

Doing the appropriate algebra, I manipulated the above expression into (a^2 - 5)^2=23

Expanding the left side, we get a^4 - 10*a^2 + 25 = 23 , i.e. a^4 - 10*a^2 + 2 = 0

So I plan to use f(x)=x^4 - 10*x^2 + 2

From here, I just need to show that it's irreducible.

If it is reducible, there will be either a linear factor or a quadratic factor.

My last step was simply to just use brute force to find a contradiction when comparing the following expressions with my polynomial above:

(ax+b)(cx^3 + dx^2 + ex + f) and

(ax^2+bx+c)(dx^2+ex+f)

Your minimal polynomial has to be monic, lets call it f(x).

f(x) has to have ##\sqrt{ 5 + \sqrt{23} }## as a root.

You also have to show there are no monic polynomials of smaller degree which have ##\sqrt{ 5 + \sqrt{23} }## as a root.

Your polynomial f(x) is indeed monic and has ##\sqrt{ 5 + \sqrt{23} }## as a root. The last part is not about showing that f(x) is irreducible ( Because it is, it has 4 linear factors ), but you want to show there is no g(x) with deg(g(x)) ≤ 3 such that g(x) is the minimal polynomial.

Last edited:
Dick