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Therefore, if H denotes the free group on X, the inversion height of the embedding of the group algebra k[H] into the Malcev-Neumann series ring is also infinite. This answer in the affirmative a question posed by Neumann in 1949 [27, p. 215]. We also give an infinite family of examples of non-isomorphic fields of fractions of k<X> with infinite invers"},"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":"1303.5287","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-03-21T15:08:00Z","cross_cats_sorted":[],"title_canon_sha256":"ba41efc27950b56328189547ff047e3af00b409333aa40d7549bfdf4009768f4","abstract_canon_sha256":"e8f3f15808799223d977b1d4ecce01db403b51dea2c992b215e0b237c03c7a56"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:13.290760Z","signature_b64":"rI4NFFstO1EDGtkYnuxifUyxVx1Ariv/rFV2EAOJ1i49py+sR/yJbz0eH2wyl2M7+1ihqx4+WK7/m54qqvaiDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"decdf45393f2b1f010974afa966bedd8fb3af14c3377f6867d5400e4ac1f5bcb","last_reissued_at":"2026-05-18T03:30:13.289868Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:13.289868Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Inversion Height of the Free Field is Infinite","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Dolors Herbera, Javier S\\'anchez","submitted_at":"2013-03-21T15:08:00Z","abstract_excerpt":"Let X be a finite set with at least two elements, and let k be any commutative field. 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