Extension of a field Watch

Dagnabbit
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#1
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#1


Part b is what I'm struggling with. Does anyone have any idea?

I know this much:

\alpha^4 + \alpha + 1
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DFranklin
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#2
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Always rusty on this, but as I recall g has to be the minimal poly for \alpha in F2. So alpha can't be the root of any poly in F2 of degree < 4.
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Dagnabbit
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(Original post by DFranklin)
Always rusty on this, but as I recall g has to be the minimal poly for \alpha in F2. So alpha can't be the root of any poly in F2 of degree < 4.
Sorry, I don't quite follow you
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Zhen Lin
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(Original post by Dagnabbit)
Sorry, I don't quite follow you
For example, if \alpha^3 = 1 then \alpha^3 - 1 = 0.
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Dagnabbit
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(Original post by Zhen Lin)
For example, if \alpha^3 = 1 then \alpha^3 - 1 = 0.
What would be so wrong with \alpha^3 - 1 = 0 ?

I've looked into this extensively and I can't spot why this is faulty
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DFranklin
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Because that contradicts X^4 + X + 1 being the minimal polynomial for alpha.
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Dagnabbit
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Okay, but the same argument cannot be used for X^5 - 1 because that is a higher order polynomial!
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SimonM
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But then (X^4+X+1) would have to divide X^5-1...
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Zhen Lin
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Alternatively, note that \alpha \ne 1, but X^5 - 1 = (X - 1)(X^4 + X^3 + X^2 + X + 1).
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