<jats:title>Abstract</jats:title><jats:p>This paper proves two results on the field of rationality<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X14007428_inline1" /><jats:tex-math>$\mathbb{Q}({\it\pi})$</jats:tex-math></jats:alternatives></jats:inline-formula>for an automorphic representation<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X14007428_inline2" /><jats:tex-math>${\it\pi}$</jats:tex-math></jats:alternatives></jats:inline-formula>, which is the subfield of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X14007428_inline3" /><jats:tex-math>$\mathbb{C}$</jats:tex-math></jats:alternatives></jats:inline-formula>fixed under the subgroup of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X14007428_inline4" /><jats:tex-math>$\text{Aut}(\mathbb{C})$</jats:tex-math></jats:alternatives></jats:inline-formula>stabilizing the isomorphism class of the finite part of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X14007428_inline5" /><jats:tex-math>${\it\pi}$</jats:tex-math></jats:alternatives></jats:inline-formula>. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representations<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X14007428_inline6" /><jats:tex-math>${\it\pi}$</jats:tex-math></jats:alternatives></jats:inline-formula>such that<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X14007428_inline7" /><jats:tex-math>${\it\pi}$</jats:tex-math></jats:alternatives></jats:inline-formula>is unramified away from a fixed finite set of places,<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X14007428_inline8" /><jats:tex-math>${\it\pi}_{\infty }$</jats:tex-math></jats:alternatives></jats:inline-formula>has a fixed infinitesimal character, and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X14007428_inline9" /><jats:tex-math>$[\mathbb{Q}({\it\pi}):\mathbb{Q}]$</jats:tex-math></jats:alternatives></jats:inline-formula>is bounded. The second main result is that for classical groups,<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X14007428_inline10" /><jats:tex-math>$[\mathbb{Q}({\it\pi}):\mathbb{Q}]$</jats:tex-math></jats:alternatives></jats:inline-formula>grows to infinity in a family of automorphic representations in level aspect whose infinite components are discrete series in a fixed<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X14007428_inline11" /><jats:tex-math>$L$</jats:tex-math></jats:alternatives></jats:inline-formula>-packet under mild conditions.</jats:p>