# Thread: [SOLVED] Algebra-How do i show this is irreducible?

1. ## [SOLVED] Algebra-How do i show this is irreducible?

My question asks me to find the minimum polynomial of .... i + root5

I get x^4 - 8x^2 + 36 = 0 as the minimum polynomial

How can i show this is irreducible.
I've tried the rational root test but that would take too long.
There must be another shorter way to do it.
Thanks.

3. To show irreducibility, observe that $\displaystyle x^{4}-8x^{2}+36=\left( x^{2}-4 \right)^2+20>0.$

And do not bump.

4. I dont understand lol how can i show irreducibility from that? Thanks.

And where does that > 0 inequality come from?

5. If i let y=x^2 so that the polynomial becomes y^2-8y+36=0, then take the discriminant which gives a negative discriminant, hence the quadratic is irreducible, would that also imply that the quartic is irreducible?

6. Originally Posted by thegarden
If i let y=x^2 so that the polynomial becomes y^2-8y+36=0, then take the discriminant which gives a negative discriminant, hence the quadratic is irreducible, would that also imply that the quartic is irreducible?
If you're all working over the real numbers, then you're correct.

~~~~~~~~~~~~~~~~~~~~~~~~~~

As for $\displaystyle (x^2-4)^2+20$, we know that a square number is always positive.

hence $\displaystyle (x^2-4)^2+20 \geq 0+20 > 0$

therefore it is never equal to 0 (for real values) and thus it is not reducible.

7. Is this the same for rational numbers?
I want to show it is irreducible in Q

8. Originally Posted by thegarden
Is this the same for rational numbers?
rational numbers are real numbers.

the thing that would change is if you're working over complex numbers (in the form a+ib)

9. Originally Posted by Moo
rational numbers are real numbers.

the thing that would change is if you're working over complex numbers (in the form a+ib)
Of course lol cheers

10. Originally Posted by Krizalid
To show irreducibility, observe that $\displaystyle x^{4}-8x^{2}+36=\left( x^{2}-4 \right)^2+20>0.$

And do not bump.
Originally Posted by Moo
If you're all working over the real numbers, then you're correct.

~~~~~~~~~~~~~~~~~~~~~~~~~~

As for $\displaystyle (x^2-4)^2+20$, we know that a square number is always positive.

hence $\displaystyle (x^2-4)^2+20 \geq 0+20 > 0$

therefore it is never equal to 0 (for real values) and thus it is not reducible.
Irreduciblity does not mean not having any zeros.
This does not fully complete the proof.

11. Originally Posted by thegarden
My question asks me to find the minimum polynomial of .... i + root5

I get x^4 - 8x^2 + 36 = 0 as the minimum polynomial

How can i show this is irreducible.
I've tried the rational root test but that would take too long.
There must be another shorter way to do it.