Characterize all algebraic number fields that have some power basis

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $K/\mathbb Q$ be a finite field extension of degree $n=[K:\mathbb Q]$. A power basis of the field means a $\mathbb Q$-vector-space basis $1,\alpha,\ldots,\alpha^{n-1}$ for some $\alpha\in K$. Equivalently $K=\mathbb Q(\alpha)$. The primitive-element theorem gives such a basis for every number field. A power integral basis would instead be a $\mathbb Z$-basis of the full ring of integers $\mathcal O_K$; that is a different condition.

Statement recorded in the supplied source.

Characterize the number fields $K$ which admit a power basis in the displayed field sense. Under this definition the characterization is all finite extensions of $\mathbb Q$, by the primitive-element theorem. [S2]

Short English statement

Which number fields have a basis given by powers of one element? With the field-basis definition in the cited source, every number field does.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400059 (AMR-093-0059). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, Algebraic number field, revision 1367779395 (2026) Bases for number fields, Power basis. https://en.wikipedia.org/w/index.php?title=Algebraic+number+field&oldid=1367779395.
• S3 Yara Elias, Kristin Lauter, Ekin Ozman and Katherine Stange, Ring-LWE Cryptography for the Number Theorist, arXiv1508.01375 (2015) Section6.3. https://arxiv.org/pdf/1508.01375.