Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 18H / b

Solution

The products

x y

with

x \in L_{1}

and

y \in L_{2}

L_{1} L_{2}

over

K

, so

[L_{1} L_{2} : K] \leq d_{1} d_{2} .

The tower law gives

[L_{1} L_{2} : K] = d_{1} [L_{1} L_{2} : L_{1}] = d_{2} [L_{1} L_{2} : L_{2}] .

Thus the degree is divisible by both

d_{1}

and

d_{2}

. Since they are coprime, it is divisible by

d_{1} d_{2}

. Combining divisibility with the upper bound gives

[L_{1} L_{2} : K] = d_{1} d_{2} .

Equivalently, finite extensions of coprime degrees are linearly disjoint field extensions.

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