Algebraic Number Theory by Frazer Jarvis

Author:Frazer Jarvis
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Corollary 6.17

Suppose that is a reduced binary quadratic form. Then if , there are automorphs of , given by ; if , there are automorphs of , given by ; in all other cases, there are just automorphs, given by .

Proof

Suppose first that . Then Lemma 6.15 shows that the only pairs with are given by . Thus a matrix which is an automorph must map into . So the first column of must be . As has determinant , this shows that for some ; however, can only be the same as if (by looking at the coefficient of say). Thus .

Next, consider the cases where . If , then Lemma 6.15 gives 4 pairs with ; these are clearly and . Then we argue as above, to conclude that (and for the same reasons as when ) or that . In this latter case, , and if , then

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