Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 18H / a

Solution

The composite

L_{1} L_{2}

is the smallest subfield of

M

containing both

L_{1}

and

L_{2}

. Since

L_{1} / K

is separable and normal, its minimal polynomials remain separable and split after base change to

L_{2}

. Hence

L_{1} L_{2} / L_{2}

is Galois.

Restriction defines

Gal (L_{1} L_{2} / L_{2}) ⟶ Gal (L_{1} / K), σ ⟼ σ ∣_{L_{1}} .

It is a homomorphism because every element of the source fixes

L_{2}

and therefore

K

. Its kernel fixes both generating fields

L_{1}

and

L_{2}

, hence fixes their composite; the map is injective.

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