Prove if&nbsp;F(an)&nbsp;is unramified or totally ramified in certain conditions

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siliciooy0j

2022-03-31

Prove if $F (\sqrt[n]{a})$ is unramified or totally ramified in certain conditions

Abdullah Avery

Beginner2022-04-01Added 19 answers

Step 1
For (1)

x^{n} - a

is separable in the residue field

\frac{O_{F}}{(π_{F})}

\frac{F (a^{\frac{1}{n}})}{F}

is automatically unramified.
Note that Hensel lemma gives that

ζ_{q - 1} \in F (a^{\frac{1}{n}})

where q is the cardinality of the residue field, and

x^{n} - a

is separable in the residue field so Hensel lemma again gives that

a^{\frac{1}{n}} \in F (ζ_{q - 1})

and

F (a^{\frac{1}{n}}) = F (ζ_{q - 1})

.
For (2) take

n l + m v (a) = 1

, let

b = a^{m} π_{F}^{n l}, v (b) = 1, F (a^{\frac{1}{n}}) = F (b^{\frac{1}{n}})

and

x^{n} - b

is Eisenstein over

O_{F}

\frac{F (b^{\frac{1}{n}})}{F}

has degree n and

v (b^{\frac{1}{n}}) = \frac{1}{n}

, it is totally ramified.

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