MHB Solve Part (b) of Nonzero Polynomials | Help Needed

Thread starter Joe20
Start date Mar 16, 2018
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Polynomials

Mar 16, 2018

Joe20

Hi all, I have solved part (a) which require verification if it is correct.

However, for part b, I am not sure how to do. Appreciate your help. Thank you.

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Mar 16, 2018

Ackbach

Gold Member

MHB

Hint: fundamental theorem of algebra, I think. What would such a polynomial have to look like?

Thread 'About the existence of Hamel basis for vector spaces'

It is well known that a vector space always admits an algebraic (Hamel) basis. This is a theorem that follows from Zorn's lemma based on the Axiom of Choice (AC). Now consider any specific instance of vector space. Since the AC axiom may or may not be included in the underlying set theory, might there be examples of vector spaces in which an Hamel basis actually doesn't exist ?

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$Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'$

Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...

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MHB Solve Part (b) of Nonzero Polynomials | Help Needed

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Thread 'About the existence of Hamel basis for vector spaces'

Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

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