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In this infinite parametric family of Galois octic fields we construct an integral basis and show non-monogenity using only congruence considerations.\n  Our method yields a new approach to consider monogenity or to prove non-monogenity in algebraic number fields. It is well applicable in parametric families of number fields. We calculate the index of elements as po"},"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":"1809.10407","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-27T08:42:43Z","cross_cats_sorted":[],"title_canon_sha256":"e71d5e714c093168a9fe4107ceaf0560c97122ee73beae3cbee904817d43da2a","abstract_canon_sha256":"432a5bb17e8e5490280491d62420f7fb92256405bb094164a06b7a6a4314c84d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:38.416944Z","signature_b64":"sTT6vmLTt4jhMNezxLPKs3TVoFj4tPpelYRRMQCcdFfbDg1EWpQasOexkfxdK4jpwqX2PPL4Pm7Ut21wDDugDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22b38f3ed37cc1ce3705b40f3ef31091d5461379610ba47ee2c27377bb2d8752","last_reissued_at":"2026-05-18T00:04:38.416501Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:38.416501Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Non-monogenity in a family of octic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Istv\\'an Ga\\'al, L\\'aszl\\'o Remete","submitted_at":"2018-09-27T08:42:43Z","abstract_excerpt":"Let $m$ be a square-free positive integer, $m\\equiv 2,3 \\; (\\bmod \\; 4)$. We show that the number field $K=Q(i,\\sqrt[4]{m})$ is non-monogene, that is it does not admit any power integral bases of type $\\{1,\\alpha,\\ldots,\\alpha^7\\}$. In this infinite parametric family of Galois octic fields we construct an integral basis and show non-monogenity using only congruence considerations.\n  Our method yields a new approach to consider monogenity or to prove non-monogenity in algebraic number fields. It is well applicable in parametric families of number fields. 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