# Residue field of composite of two field

[Question]

I know that $$K'\cdot K''$$ is an unramified extension of $$K$$ but I don't know why $$K'\cdot K''$$ have a residue field $$k'$$.

is it always true that $$K_1\cdot K_2$$ have a residue field $$k_1 \cdot k_2$$? (where $$k_1,k_2$$ are residue fields of $$K_1, K_2$$)

I think that if we prove the proposition 7.50 , then we can use " $$K_1\cdot K_2$$ have a residue field $$k_1 \cdot k_2$$" in this situation.

However, we can't use that fact while proving this proposition.

How can I prove this?

reference(J.S. Milne's Algebraic Number Theory) and this post 1: Strange reasoning of unramified extensions having same residue fields are the same.

For $$K/\Bbb{Q}_p$$ a finite extension then $$F/K$$ is unramified iff $$F=K(\zeta_n)=K(\zeta_{q-1})$$ with $$p\nmid n$$ and $$q= |O_F/(\pi_F)|$$. This is the main application of Hensel lemma.

When $$E/K,E'/K$$ are ramified then it is not always the case that the residue field of $$EE'$$ is the smallest field contained those of $$E,E'$$, try with $$E=\Bbb{Q}_2(2^{1/3}),E'=\Bbb{Q}_2(\zeta_3 2^{1/3})$$.

When $$E'/K$$ is unramified then $$EE'=E(\zeta_{q-1})$$ has residue field $$O_E/(\pi_E)(\zeta_{q-1})$$.

• Thank you. May I ask you something? If the base field is $\mathbb{F}_p((t))$ ( not $\mathbb{Q}_p$, then still this hold? (Maybe, you assumed that characteristic is $0$)
– hew
Jan 4, 2021 at 12:44
• Also, is $n$ any natural number and $q$ prime number such that $p\nmid n$ and $q= |O_F/(\pi_F)|$ ? Then, you mean, there are infinitely many $n$ such that $F=K(\zeta_n)$. Probably, it seems to be misunderstood. So, may I ask you the reference with proof of this "the main application of Hensel lemma" ? Finally, I don't understand why this imply that $K'\cdot K''$ have a residue field $k'$. Sorry and thank you.
– hew
Jan 4, 2021 at 12:54
• It works the same way for finite extensions of $\Bbb{F}_p((t))$ and any complete discretely valued fields. Unramified means that $[F:K]=[O_F/(\pi_F):O_K/(\pi_K)]$, take a generator of the residue field extension, lift its minimal polynomial to $O_K[x]$ which stays irreducible, Hensel lemma gives a root $a$ of it in $O_F$ then $[K(a):K]=[O_F/(\pi_F):O_K/(\pi_K)]$ thus $K(a)=F$. Jan 4, 2021 at 13:56
• $\zeta_{q-1}$ is when the residue field is finite. Jan 4, 2021 at 13:56