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#1 2009-01-24 02:50:30
- gepolv
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- Registered: 2009-01-24
- Posts: 2
the concept of "a polynomial over Galois Field"
Hi, all,
I am not quite understanding the concept of "a polynomial over Galois Field"? whether "over some field" means the coefficients of the polynomial should be in some field? For example, make 4x^2+5x+6 over GF(4). Whether this means the coefficients 4,5,6 should be changed in GF(4). Here, we should notice that GF(4) is not Z_4.
thank you.
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#2 2009-01-24 03:46:58
- gepolv
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- Registered: 2009-01-24
- Posts: 2
Re: the concept of "a polynomial over Galois Field"
That is not right. Z_4 includes {0,1,2,3}, but for GF(4), that is not correct.
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#3 2009-01-26 07:26:06
- mathsmypassion
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- Registered: 2008-12-01
- Posts: 33
Re: the concept of "a polynomial over Galois Field"
GF(p) represents a Galois field when p is prime. In our problem p = 4 that is not a prime number.
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#4 2009-01-26 11:05:13
- Ricky
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- Registered: 2005-12-04
- Posts: 3,791
Re: the concept of "a polynomial over Galois Field"
GF(p) represents a Galois field when p is prime. In our problem p = 4 that is not a prime number.
No, GF(p^e) represents the Galois field of prime power order, and as such 2^2 = 4 is perfectly valid.
To say that 4x^2+5x+6 is over GF(4) does not make sense. To represent GF(4), you need to label one element alpha, the root of a 2nd order irreducible polynomial over Z_2. I suppose you could possibly reduce them down by Z_2, but that would just leave x.
But yes, in general, a polynomial over a field F means that the coefficients of that polynomial come from F.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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#5 2009-01-26 12:04:34
- JaneFairfax
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- Registered: 2007-02-23
- Posts: 6,868
Re: the concept of "a polynomial over Galois Field"
To say that 4x^2+5x+6 is over GF(4) does not make sense.
That was what I tried to indicate to the OP, but the OP wouldnt listen. Hence I deleted my post and left gepolv to talk to him-/herself.
Lets see if gepolv will listen to you instead. 
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