4.3. Polynomials of degree ≤ 3.

Back

Example 4.5. Let f(X) ∈ F[X] be a polynomial of degree 2. Then f is inseparable

⇐⇒ F has characteristic 2 and f(X) = X2 −a for some a ∈ F \ F2. If f is separable, then

Gf = 1(= A2) or S2 according as D(f) is a square in F or not.

Example 4.6. Let f(X) ∈ F[X] be a polynomial of degree 3. We can assume f to be

irreducible, for otherwise we are essentially back in the previous case. Then f is inseparable

⇐⇒ F has characteristic 3 and f(X) = X3 −a some a ∈ F \ F3. If f is separable, then Gf

is a transitive subgroup of S3 whose order is divisible by 3. There are only two possibilities:

Gf = A3(=< (123) >) or S3 according as D(f) is a square in F or not.

For example, X3−3X+1 ∈ Q[X] is irreducible (apply 1.4), its discriminant is −4(−3)3−

27 = 81 = 92, and so its Galois group is A3.

On the other hand, X3+3X+1 ∈ Q[X] is also irreducible (apply 1.4), but its discriminant

is −135 which is not a square in Q, and so its Galois group is S3.

Навигация