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Are the following irreducible over Q? | Are the following irreducible over Q? | ||
| − | a) <math>x^5 + 9x^4 + 12x^2 + 6</math> | + | *a) <math>x^5 + 9x^4 + 12x^2 + 6</math> |
| − | b) <math>x^4 + x + 1</math> | + | *b) <math>x^4 + x + 1</math> |
| − | c) <math>x^4 + 3x^2 + 3</math> | + | *c) <math>x^4 + 3x^2 + 3</math> |
| − | d) <math>x^5 + 5x^2 + 1</math> | + | *d) <math>x^5 + 5x^2 + 1</math> |
| − | e) <math>(5/2)x^5 + (9/2)x^4 + 15x^3 + (3/7)x^2 + 6x + (3/14)</math> | + | *e) <math>(5/2)x^5 + (9/2)x^4 + 15x^3 + (3/7)x^2 + 6x + (3/14)</math> |
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Revision as of 17:25, 8 April 2009
Are the following irreducible over Q?
- a) $ x^5 + 9x^4 + 12x^2 + 6 $
- b) $ x^4 + x + 1 $
- c) $ x^4 + 3x^2 + 3 $
- d) $ x^5 + 5x^2 + 1 $
- e) $ (5/2)x^5 + (9/2)x^4 + 15x^3 + (3/7)x^2 + 6x + (3/14) $
a.) Look at Eisenstein's with p = 3.
b.) A polynomial is irreducible in Q if there's a p such that f(x) mod p is irreducible. Look at p = 2.
c.) See part a.
d.) See part b.
e.) Multiply by 14 then see part a.
--Jniederh 22:12, 8 April 2009 (UTC)
