Normal bases of small height in Galois number fields
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910017329823744 |
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| author | Fukshansky, Lenny Jeong, Sehun |
| author_facet | Fukshansky, Lenny Jeong, Sehun |
| contents | Let $K$ be a number field of degree $d$ so that $K/\mathbb Q$ is a Galois extension. The {\it normal basis theorem} states that $K$ has a $\mathbb Q$-basis consisting of algebraic conjugates, in fact $K$ contains infinitely many such bases. We prove an effective version of this theorem, obtaining a normal basis for $K/\mathbb Q$ of bounded Weil height with an explicit bound in terms of the degree and discriminant of $K$. In the case when $d$ is prime, we obtain a particularly good bound using a different method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04437 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Normal bases of small height in Galois number fields Fukshansky, Lenny Jeong, Sehun Number Theory 11G50, 11R04, 11R32, 11H06 Let $K$ be a number field of degree $d$ so that $K/\mathbb Q$ is a Galois extension. The {\it normal basis theorem} states that $K$ has a $\mathbb Q$-basis consisting of algebraic conjugates, in fact $K$ contains infinitely many such bases. We prove an effective version of this theorem, obtaining a normal basis for $K/\mathbb Q$ of bounded Weil height with an explicit bound in terms of the degree and discriminant of $K$. In the case when $d$ is prime, we obtain a particularly good bound using a different method. |
| title | Normal bases of small height in Galois number fields |
| topic | Number Theory 11G50, 11R04, 11R32, 11H06 |
| url | https://arxiv.org/abs/2601.04437 |