Show that if F is a finite extension of Q, then the torsion subgroup of F*...

Show that if F is a finite extension of Q, then the torsion subgroup of F^* is finite

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Expert Solution

Claim to be proved: The number of elements of the torsion subgroup of F* is finite.

For any field F the group of units (invertible elements) is precisely F*. So to understand the torsion subgroup of F*, we only need to know the set of roots of unity in the group F. [This is because of the definition of torsion subgroup of F*]

Now it is given that F is a finite extension of Q which means the index [F:Q] is a finite number say 'n'.

If F contains an k^th root of unity then it must also contain a primitive root say a d^th root. so the numbers k and d are such that 'd' divides 'k'.

So it suffices for us to show that F contains only finite number of primitive roots. Fixing a primitive d^th root of unity in F one observes that the Euler totient function phi(d) divides 'n'. Clearly the number of such d's is finite since the number of divisors of 'n' is clearly finite for any 'n'.

Hence the claim that the torsion subgroup of F* is Finite

milcah answered 1 year ago

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