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Factorization of Polynomials - Irreducibles - Anderson and Feil

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
I am reading Anderson and Feil - A First Course in Abstract Algebra.

On page 56 (see attached) ANderson and Feil show that the polynomial [TEX] f = x^2 + 2 [/TEX] is irreducible in [TEX] \mathbb{Q} [x][/TEX]

After this they challenge the reader with the following exercise:

Show that [TEX] x^4 + 2 [/TEX] is irreducible in [TEX] \mathbb{Q} [x][/TEX]. taking your lead from the discussion of [TEX] x^2 + 2 [/TEX] above. (see attached)

Can anyone help me to show this in the manner requested. Would appreciate the help.

Peter
 

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
I am reading Anderson and Feil - A First Course in Abstract Algebra.

On page 56 (see attached) ANderson and Feil show that the polynomial [TEX] f = x^2 + 2 [/TEX] is irreducible in [TEX] \mathbb{Q} [x][/TEX]

After this they challenge the reader with the following exercise:

Show that [TEX] x^4 + 2 [/TEX] is irreducible in [TEX] \mathbb{Q} [x][/TEX]. taking your lead from the discussion of [TEX] x^2 + 2 [/TEX] above. (see attached)

Can anyone help me to show this in the manner requested. Would appreciate the help.

Peter
Hi Peter, :)

I don't see any attachments in your post. You can use a image hosting website such as TinyPic to upload images and link them here, if you have trouble attaching files.

To show that \(x^4 + 2\) is irreducible over \(\mathbb{Q}[x]\) you can use Eisenstein's Irreducibility Criterion.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
Hi Peter, :)

I don't see any attachments in your post. You can use a image hosting website such as TinyPic to upload images and link them here, if you have trouble attaching files.

To show that \(x^4 + 2\) is irreducible over \(\mathbb{Q}[x]\) you can use Eisenstein's Irreducibility Criterion.


Thanks - most helpful - appreciate your help

The reason I did not upload the attachement was that I could not delete my old attachements - about 5 or so are there and they exceed my allowed quota _ I cannot seem to delete them

Peter
 

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
Thanks - most helpful - appreciate your help

The reason I did not upload the attachement was that I could not delete my old attachements - about 5 or so are there and they exceed my allowed quota _ I cannot seem to delete them

Peter
To if you want to delete your previous attachments go to "Settings" and then click on the "Attachments" under the "My Settings" pane. Hope this will work for you. :)