## 541 Reputation

15 years, 192 days

ok, I got it.

Thank you.

## That  is interesting. but...

That  is interesting.

but how the groebner basis works for the undertermined root.

for example:

alias(alpha=RootOf(Z^3+Z+1));
F := {alpha*x*y+alpha^2+1, alpha^2*x^2-1};
G := Groebner[Basis](F, plex(x,y), characteristic=2);

## Yes, the problem is how to...

Yes, the problem is how to compare a real number and the root?

Thanks, Alex.

## Thank you so much. I am...

Thank you so much. I am trying to understand your procedure.

Thanks again.

## from the following, it said...

from the following, it said

GF(4):=GF(2)/x^2+x+1;

So we have a field with 22 = 4 elements, namely one representation of $\mathbb{F}_4$. The fields elements are 0, 1, x, and x+1 (choosing their cannonical representatives). In the ring of integers, the elements 0,1,2 and 3 are chosen to represent the field. To map between both sets the substitution map, substituting x with 2 is chosen, so we have

```0 <-> 0
1 <-> 1
2 <-> x
3 <-> x+1.
```

A polynomial Ring Z[y[1..3]]; can be used to represent $\mathbb{F}_4[y_1, y_2, y_3]$. Therefore, we map Z[y[1..3]]; to Z/(4)[1..3]]; and interpret each integer via the above described map. So the polynomial 3y[1] + 1y[2] + 5y[3] - 1 y[1]y[2]; is first mapped to 3y[1] + 1y[2] + 1y[3] +3 y[1]y[2]; which means the polynomial (x + 1)y1 + y2 + y3 + (x1)y1y2.

------------------------------------------------------------------------------

Is that right? Make all the elements in GF(4) correspond with a integer number from 0 through 3.

Thank you.

## GB([x^2-2*x*z+5,...

```GB([x^2-2*x*z+5, x*y^2+y*z^3, 3*y^2-8*z^3],4);

3      8    9     3      7    8      5   2      3
[-7 z  - 4 z  + z , y z  + 8 z  - z  + 3 z , y  - 7 z ,

3      7      8      5   2
x z  + 3 z  - 3 z  + 6 z , x  - 2 x z + 5]

GB({x^2+2*y, x^3+5},4);

3         2
[-3 + y , x - 6 y ]
```

Alec

---------------------------------------------------------------------------

I cannot understand why the coefficients are not in GF(16);

Thank you.

## and for the following...

and for the following example, how to compute the groebner basis?

F := [x^2-2*x*z+5, x*y^2+y*z^3, 3*y^2-8*z^3] over GF(2^4)

which is different from your example: the coefficients are all integers, not root of some irreducible polynomials.

```alias(a=RootOf(_Z^4+_Z+1)):
Groebner:-Basis({x^2+a*y,x^3+(a^2+1)},
tdeg(y,x),characteristic=2);```

Thank you.

## I am not quite clear...

I am not quite clear that.
for example, for a polynomial "5x^2+3y^2" over a field GF(2^4), all the coefficients should be transformed into the number in Field GF(16), i.e., coefficients (5 and 3) should be changed to the number in GF(16)? if so, how to change?

## 1.Well, I understand how to...

1.Well, I understand how to compute Groebner Basis over R, or GF(p), p is a prime number and actually I have implemented the B's algorithm and made the result of B's algorithm unique. The problem now is I am not quite clear how to compute Groebner Basis over GF(2^m).

2,Do you know the differences between ideal over R(real number) and ideal over R[x]?

Thank you

## I benefit a lot from your...

BY the way, do you know how to computer the Groebner Basis over GF(2^m) manually?  or could you give me a link about that?

## Yes, I also tried that. I...

Yes, I also tried that. I met the same problems.

Maybe there are other ways to issue it.

Thank you, Alex. I really appreciate your help.

## Well, the problem is I have...

Well, the problem is I have already completed part of the programming. So now it is difficult for me to transfer my codes to sage or other platform.

If maple cannot offer such operations, I will try to implement it in maple.  But now I have some problems.

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 so much. But What...

Thank you so much. But What i want to do is to do some programming using maple.

Any other ideas?

thanks a lot.

## to  Robert Israel: yes,...

yes, when you want to newline, shift+enter instead of enter can get:

> F:=proc(L)

>local LL;

>  LL:=subsop(1=NULL,LL);
> end proc:

> L:=[x,y,z,d];
L := [x, y, z, d]

>L:= F(L);
[y, z, d]
>L:= F(L);
[z, d]
>L:= F(L);
[d]
>F(L);
[]

actually, it is the same as:

F := proc(L) local LL; LL:=subsop(1=NULL,LL);(wihtout "enter")end proc;

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