Finding the Galois Field of a number

[2FA]

This came up for me today on my encryption exam.

 

There was a question stating "Draw the multiplicative inverse table of GF(11)" which I understand. On a homework, there was a question that only stated "Find GF(17)" which I don't understand.

 

Could someone explain finding the Galois Field of a number, because my professor didn't really explain it to begin with. She just showed us the generic form in terms of X and then did a couple of examples of the tables.

[SteveGrabowski0]

1 hour ago, DeadEyePsycho said:

This came up for me today on my encryption exam.

 

There was a question stating "Draw the multiplicative inverse table of GF(11)" which I understand. On a homework, there was a question that only stated "Find GF(17)" which I don't understand.

 

Could someone explain finding the Galois Field of a number, because my professor didn't really explain it to begin with. She just showed us the generic form in terms of X and then did a couple of examples of the tables.

The only finite fields are of the form GF(p^n) = Z/p^nZ where Z is the set of all integers, n is a positive integer, and p is a prime number. So GF(17) is just isomorphic to Z/17Z, eg, the set of integers modulo 17 with their associated addition, multiplication, and division operations. So it's just the integers {0,1,2,...,16} with the associated addition, multiplication, and division operations in Z/17Z. Eg the addition operation + goes by a + b = a + b mod 17, eg, the remainder of the integer a+b when divided by 17 and similar for multiplication. The only part of the problem that's nontrivial is finding the division operation, which is just finding the inverse table like you said above. Eg like 4^{-1} = 13 since 4*13 =  1 + 3*17 so that 4*13 = 1 mod 17.

[2FA]

51 minutes ago, SteveGrabowski0 said:

The only finite fields are of the form GF(p^n) = Z/p^nZ where Z is the set of all integers, n is a positive integer, and p is a prime number. So GF(17) is just isomorphic to Z/17Z, eg, the set of integers modulo 17 with their associated addition, multiplication, and division operations. So it's just the integers {0,1,2,...,16} with the associated addition, multiplication, and division operations in Z/17Z. Eg the addition operation + goes by a + b = a + b mod 17, eg, the remainder of the integer a+b when divided by 17 and similar for multiplication. The only part of the problem that's nontrivial is finding the division operation, which is just finding the inverse table like you said above. Eg like 4^{-1} = 13 since 4*13 =  1 + 3*17 so that 4*13 = 1 mod 17.

 

Thanks for clearing that up. Like I said, it was never really explained that's what the question would be asked like, we were just shown the tables so it kind of threw me off.

[SteveGrabowski0]

I had a professor like that in freshman physics who would show something really quickly and just act like everything was obvious from it.

[2FA]

Just now, SteveGrabowski0 said:

I had a professor like that in freshman physics who would show something really quickly and just act like everything was obvious from it.

You know what's funny is that the entire physics department here is like that.