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For these groups, we produce two extensions $L/\\Qq(U)$ that cannot be simultaneously induced, thus even disproving a weaker Lifting Property. Our examples of such groups $G$ include symmetric groups $S_n$, $n\\geq 7$, infinitely many $PSL_2(\\Ff_p)$, the Monster. Two variants of the question with $\\Qq(U)$ replaced by $\\Cc(U)$ and $\\Qq$ are answered similarly, t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1605.09363","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-30T19:42:06Z","cross_cats_sorted":[],"title_canon_sha256":"d224e27a0db4002b29ecf934639196660fd3388a5a7887910a65edecdb43b056","abstract_canon_sha256":"87a7fb4f7e9665d93549c3c8e47aabfc24d6489a29ce6f91499b4e6133efe50c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:21.302340Z","signature_b64":"M6VhmcFnT6deh0F0AC94GMBsA0rIdSnY/rtI89aStyDy0RNwtq07jV6mpL39vW29Wfbyn1+1WH9CKzmaZNxSBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0bece6e9a66e7f2e93f35a0489f847967d5d498b6543bde9e1426fabd088673b","last_reissued_at":"2026-05-18T01:13:21.301744Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:21.301744Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Groups with no Parametric Galois Extension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Pierre D\\`ebes","submitted_at":"2016-05-30T19:42:06Z","abstract_excerpt":"We disprove a strong form of the Regular Inverse Galois Problem: there exist finite groups $G$ which do not have a realization $F/\\Qq(T)$ that induces all Galois extensions $L/\\Qq(U)$ of group $G$ by specializing $T$ to $f(U) \\in \\Qq(U)$. For these groups, we produce two extensions $L/\\Qq(U)$ that cannot be simultaneously induced, thus even disproving a weaker Lifting Property. Our examples of such groups $G$ include symmetric groups $S_n$, $n\\geq 7$, infinitely many $PSL_2(\\Ff_p)$, the Monster. 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