Galois Fields

drkirkby · 2022-12-24, 22:42

Someone asked the following on an amateur radio forum. I would be interested in any answers, that I can point him to. He is a very switched-on engineer, but does not have any number theory books.

A simple question none of the mathematical texts I've tried to study have actually made clear :

Is Modulo arithmetic under MOD(any prime number) a Galois Field?
Always?
If not, what's different?
addition and multiplication work in the field, ie (a + b) MOD p and a.b
MOD p always result in complete use of the set of numbers 0 to p-1 so I
think that qualifies. Why do the books not appear to explicitly say so?

The texts discuss module prime arithmetic being a 'field', then usually
move to polynomials and Galois Fields in the next page or two. And I'm
missing something in the intermediate jump. Is it assumed and I'm being
too pedantic?

charybdis · 2022-12-24, 23:50

A Galois field is just another name for a finite field. So yes, arithmetic modulo any prime is a Galois field - though not every Galois field is of this type.

drkirkby · 2022-12-25, 06:45

Quote:

Originally Posted by charybdis

A Galois field is just another name for a finite field. So yes, arithmetic modulo any prime is a Galois field - though not every Galois field is of this type.

Thank you.

2022-12-24, 22:42	#1
drkirkby "David Kirkby" Jan 2021 Althorne, Essex, UK 2·229 Posts	Galois Fields Someone asked the following on an amateur radio forum. I would be interested in any answers, that I can point him to. He is a very switched-on engineer, but does not have any number theory books. A simple question none of the mathematical texts I've tried to study have actually made clear : Is Modulo arithmetic under MOD(any prime number) a Galois Field? Always? If not, what's different? addition and multiplication work in the field, ie (a + b) MOD p and a.b MOD p always result in complete use of the set of numbers 0 to p-1 so I think that qualifies. Why do the books not appear to explicitly say so? The texts discuss module prime arithmetic being a 'field', then usually move to polynomials and Galois Fields in the next page or two. And I'm missing something in the intermediate jump. Is it assumed and I'm being too pedantic? Last fiddled with by drkirkby on 2022-12-24 at 22:48

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2022-12-24, 23:50	#2
charybdis Apr 2020 13·79 Posts	A Galois field is just another name for a finite field. So yes, arithmetic modulo any prime is a Galois field - though not every Galois field is of this type.