Show that a polynomial of order 2 or 3 is irreducible on a field if has a root in .

Use this to show that the polynomial is irreducible in

Eisenstein's creiterio ??

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- Jun 8th 2009, 07:59 AMosodudIrreducible Polynomial 2
Show that a polynomial of order 2 or 3 is irreducible on a field if has a root in .

Use this to show that the polynomial is irreducible in

Eisenstein's creiterio ?? - Jun 8th 2009, 08:54 PMNonCommAlg
- Jun 8th 2009, 08:57 PMosodud
- Jun 8th 2009, 09:15 PMNonCommAlg
this is false. fix the mistake(s) you made and then i'll help you.

Quote:

Use this to show that the polynomial is irreducible in

Eisenstein's creiterio ??

__integer__

__solution__. suppose it has integer solutions. then and hence we must have both and which cannot happen and you're done. - Jun 8th 2009, 09:21 PMosodud
- Jun 8th 2009, 09:25 PMNonCommAlg
- Jun 8th 2009, 09:53 PMosodudFinal
Show that a polynomial of order 2 or 3 is reducible in a Field if and only if has one root in .

That is what the exercise says.

I swear¡¡

If the exercise is incorrect could you tell me why is so??

you rule

Thanks again

- Jun 8th 2009, 10:22 PMNonCommAlg
it is correct now. one side of the statement holds for any polynomial of degree at least 2: if a polynomial of degree at least 2 has a root then it has to be divisible by and thus it

is reducible. conversely, suppose is a reducible polynomial of degree 2 or 3. then it can be factored into non-constant polynomials of smaller degrees. but how can you factor a

polynomial of degree 2 into non-constant polynomials of smaller degrees? there's one way only: two polynomials of degree 1. so suppose then obviously

(and ) would be a root of now suppose that the degree of is 3. how can be factored into non-constant polynomials of smaller degrees? there are two ways: three

polynomials of degree 1 or one polynomial of degree 1 and one polynomial of degree 2. in either case has a factor of degree 1, say and thus would be a root of