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Extension of a field

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

I know this much:

\alpha^4 + \alpha + 1

Reply 1

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.

Reply 2

Dagnabbit

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

Reply 3

Zhen Lin

Original post by Dagnabbit

Sorry, I don't quite follow you

For example, if

\alpha^3 = 1

then

\alpha^3 - 1 = 0

Reply 4

Dagnabbit

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

Reply 5

DFranklin

Because that contradicts X^4 + X + 1 being the minimal polynomial for alpha.

Reply 6

Dagnabbit

Okay, but the same argument cannot be used for X^5 - 1 because that is a higher order polynomial!

Reply 7

SimonM

But then (X^4+X+1) would have to divide X^5-1...

Reply 8

Zhen Lin

Alternatively, note that

\alpha \ne 1

, but

X^5 - 1 = (X - 1)(X^4 + X^3 + X^2 + X + 1)

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Extension of a field

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