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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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---- | ---- | ||
| − | a.) Look at Eisenstein's with p = 3. | + | a.) Look at Eisenstein's with p = 3.<br> |
| − | b.) A polynomial is irreducible in Q if there's a p such that f(x) mod p is irreducible. Look at p = 2. | + | b.) A polynomial is irreducible in Q if there's a p such that f(x) mod p is irreducible. Look at p = 2.<br> |
| − | c.) See part a. | + | c.) See part a.<br> |
| − | d.) See part b. | + | d.) See part b.<br> |
| − | e.) Multiply by 14 then see part a. | + | e.) Multiply by 14 then see part a.<br> |
--[[User:Jniederh|Jniederh]] 22:12, 8 April 2009 (UTC) | --[[User:Jniederh|Jniederh]] 22:12, 8 April 2009 (UTC) | ||
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| + | ---- | ||
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| + | Einstein's explicitly says p cannot divide any <math>a_n</math> so p cannot be 3 for parts a, c or e<br> | ||
| + | --[[User:Bcaulkin|Bcaulkin]] 22:38, 8 April 2009 (UTC) | ||
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| + | ---- | ||
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| + | According to theorem 17.4 in the book says that the leading coefficient cannot be divisible by p, but all the others must be. In addition the last coefficient cannot be divisible by p^2. | ||
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| + | ---- | ||
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| + | My fault, I misread it. Thanks! | ||
Latest revision as of 17:50, 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)
Einstein's explicitly says p cannot divide any $ a_n $ so p cannot be 3 for parts a, c or e
--Bcaulkin 22:38, 8 April 2009 (UTC)
According to theorem 17.4 in the book says that the leading coefficient cannot be divisible by p, but all the others must be. In addition the last coefficient cannot be divisible by p^2.
My fault, I misread it. Thanks!
