This is a statement in Serre's A Course in Arithmetic (p. 18).
If $p\ne2$, the group $\mathbb{Q}_p^*/\mathbb{Q}_p^{*2}$ is a group of type $(2,2)$.
What is a group of type $(2,2)$?
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For a prime $p$, a group of type $(p,p)$ is one that is a direct product of two groups of order $p$. So a group of type (2,2) is a product of two groups of order 2. – KCd Apr 01 '13 at 10:56
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Yes, it is annoying when authors use undefined notation, in particular if it is not standard one: this time this seems to be closer to what Fuchs uses in his book: type $,(2,2), $ means, apparently, two direct factors of order two each, or what we poor, miserable mortals would also call the Klein group $,C_2\times C_2, $ ,with $,C_n=$ the cyclic group of order two. – DonAntonio Apr 01 '13 at 10:59
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Ah I see. Thanks a lot! – abc Apr 01 '13 at 11:00
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I expose the cyclotomic structure of ${\bf Q}_p$ in this previous answer (that is, I find the isomorphism class of ${\bf Q}_p^\times/({\bf Q}_p^\times)^n$ exactly for all primes $p$ and $n\ge1$). The simpler case ${\bf Q}_p^\times/({\bf Q}_p^\times)^2$ for odd $p$ is easier to deduce as being $C_2\times C_2$, the Klein four group. It would seem the "typing" system that Serre refers to has to do with either the $p$-primary or invariant factor decomposition of f.g. abelian groups (into cyclic groups whose orders are put into tuples), see Wikipedia for more details.
