Galois group of a product of irreducible polynomials - MathOverflow

Galois group of a product of irreducible polynomials

2010-03-26T15:35:22Z

Hello

Suppose given a polynomial $P=Q_1\cdots Q_k$ of degree $n$, where each $Q_i$ is irreducible. Suppose also that I know the Galois group $G_i$ (over the rationals) of each irreducible factor $Q_i$.

Is there an easy correlation between the Galois group of $P$, and the $G_i$?

Answer by Robin Chapman for Galois group of a product of irreducible polynomials

2010-03-26T15:44:08Z

The Galois group of $P$ will be a subdirect product of the $G_i$, that is a subgroup of $G_1\times\cdots\times G_k$ projecting surjectively onto each of the $G_i$.

Answer by Martin Bright for Galois group of a product of irreducible polynomials

2010-03-26T15:47:26Z

If you mean: does knowing the G_i tell you the Galois group of P, then no.

Examples:

$P = (X^2+1)(X^2-2)$ has Galois group $C_2 \times C_2$, and both factors have Galois group $C_2$; this works because the splitting fields of the two factors intersect only in $\mathbb{Q}$.

But $P = (X^2 + X + 1)(X^2+3)$ has Galois group $C_2$, although both factors again have Galois group $C_2$. Here both factors, though they're coprime, define the same extension $\mathbb{Q}(\sqrt{-3})$.

I've just seen Robin's answer, so to relate to that: in the first example, the Galois group of P is the whole of $G_1 \times G_2$. In the second example, it is the diagonal subgroup of $G_1 \times G_2$, which is smaller although still projects surjectively onto each factor.