Find out the answer to "Order of field extension Im"

scrimaeua

Answered question

2022-02-12

Order of field extension
Im

meldElafellrbo

Beginner2022-02-13Added 13 answers

Let

f = x^{3} + a x^{2} + b x + c

be an irreducible polynomial over

Z_{11}

.
The extension field

E = Z_{11} \frac{x}{⟨ f ⟩}

contains a zero of f, namely the residue class

α = \overset{―}{x} + ⟨ f ⟩

, This gives

α^{3} + a α^{2} + b α + c =, so α^{3} = - a α^{2} - b α - c

, and the extension field is

E = {u α^{2} + v α + w ∣ u, v, w \in Z_{11}}

with degree

[E : Z_{11}] = 3

. In particular, if f is primitive, the powers of

α

are exactly the nonzero elements of E.
NB:

f (\overset{―}{x}) = f (x + ⟨ f ⟩) = f (x) + ⟨ f ⟩ = ⟨ f ⟩ = \overset{―}{0}

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