Question about F_5

I am currently stuck on question 2 of problem sheet 3. After changing the polynomials into their mod 5 forms, do I divide them to find the hcf as normal (without changing the remainders etc into their mod 5 forms) or after every division work out the mod 5 forms of the polynomials. For the Bezout’s lemma will I work out h and k and then convert them into their mod 5 forms right at the end. Sorry if that is a little confusing. Many Thanks.

Reply:

One way is to divide normally, just remembering at each stage to set any multiple of 5 to zero. At the end of all your calculations, convert to elements of F_5[x]. For example, if your answer comes out to be

x^3-4x+(1/3)

in F_5[x] this would become

x^3+x+2

since -4=1 and (1/3)=2 in F_5. For small calculations done by humans, this is probably the easiest process, at least from a psychological standpoint. However, you should be aware that for large calculations done by machines (or by humans used to F_5) converting at each stage to the F_5 form is much more efficient.

Feel free to ask again after trying this out.

MK

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