американской

американской

"American" (adjective, feminine singular or genitive plural).

Here 'американской' is an adjective in the genitive case, agreeing with 'газеты' (newspaper), indicating possession or association.

Я прочитал интересную статью из американской газеты.

I read an interesting article from an American newspaper.

In this case, 'американской' is also an adjective in the genitive case, describing 'кухни' (cuisine) to specify its origin.

Мы заказали блюда из американской кухни.

We ordered dishes from American cuisine.

Here 'американской' is an adjective in the instrumental case, correlating with 'гражданством' (citizenship) to denote the quality or attribute belonging to it.

Она гордится своим американской гражданством.

She is proud of her American citizenship.