= Ramification in the quadratic extension generated by the square root of two
{title2=$G_1=G_2=C_2,\qquad G_i=1\ (i\ge3)$}
In $L=\mathbb Q_2(\sqrt2)$, set $\pi=\sqrt2$ and normalize $v_L(\pi)=1$, so $v_L(2)=2$. The two terms of $a+b\pi$ have valuations of opposite parity, giving $\mathcal O_L=\mathbb Z_2\oplus\mathbb Z_2\pi$. The nonidentity automorphism sends $\pi$ to $-\pi$, and $v_L(-2\pi)=3$. The <lower ramification numbering> therefore gives the displayed groups. The different exponent is three.
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