(New page: Are the following irreducible over Q? a) <math>x^5 + 9x^4 + 12x^2 + 6</math> b) <math>x^4 + x + 1</math> c) <math>x^4 + 3x^2 + 3</math> d) <math>x^5 + 5x^2 + 1</math> e) <math>(5/2)x^5 + ...) |
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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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| + | ---- | ||
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| + | 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. | ||
| + | --[[User:Jniederh|Jniederh]] 22:12, 8 April 2009 (UTC) | ||
Revision as of 18:12, 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)
