Artin Field Extension

Emile Artin’s very unique book “Galois Theory” (1971) on “Finite Field Extension” interpreted by Vector Space.

Let H a Field with subfield G
F is G’s subfield:
H ⊃ G ⊃ F

Example:
Let
F = Q = Rational Field
G = Q(√2) = Larger Extended Field Q with irrational root √2
H = Q(√2, √3) = Largest Extended Field Q with irrational roots (√2 & 3)

{1, √2} forms basis of Q(√2) over Q

{1, √3} basis of Q(√2, √3) over Q(√2)
[since √3 ≠ p+ q√2 , ∀p,q ∈ Q]

=> {1,√2, √3, √6} basis of Q(√2, √3) over Q
=> Q(√2, √3) is a 4-dimensional Vector Space over Q.

Isomorphism (≌)

Q(√2) Q[x] / {x² – 2}

Read as:

Q(√2) isomorphic to the quotient of the Polynomial ring Q[x] modulo the Principal Ideal {x² – 2}
Q[x] the Polynomial Ring
{x² – 2} is the Principal Ideal
Complex Number (C)
C = R[x] / {x² + 1}
R[x] the Polynomial Ring with coefficients in the Field R
{x² + 1} is the Principal Ideal
Questions:
Since R[x] / {x² + 1} is the Field C

Why below are not Fields ?
R[x] / {x³ + 1}
R[x] / {x^4 + 1}
C[x] / {x² + 1}

Hint: they are not irreducible in that particular Field, not a Principal Ideal.

Note: C[x] the Polynomial Ring with coefficients in the Field C

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