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Old 2022-12-24, 22:42   #1
drkirkby
 
"David Kirkby"
Jan 2021
Althorne, Essex, UK

457 Posts
Default 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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Old 2022-12-24, 23:50   #2
charybdis
 
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Apr 2020

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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.
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Old 2022-12-25, 06:45   #3
drkirkby
 
"David Kirkby"
Jan 2021
Althorne, Essex, UK

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Quote:
Originally Posted by charybdis View Post
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.
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