nLab transcendental element

Definition

Given a field $K$ and a field extension $K \subseteq F$, an element $\alpha \in F$ is transcendental if for all polynomial functions $f$ with coefficients in $K$, if $f(\alpha) = 0$ then $f$ is the zero polynomial.

Alternatively, an element $\alpha \in F$ is transcendental if the field extension $K(\alpha) \subseteq F$ is isomorphic to the field of fractions $K(x)$ of the generic polynomial ring $K[x]$.

In constructive mathematics

These two notions coincide in classical mathematics. They also coincide in constructive mathematics if $F$ and $K$ are both discrete fields. However, they are different in constructive mathematics if it is not the case that $F$ and $K$ are both discrete fields. If $K$ is only a Heyting field, the first notion is weaker and so is called weakly transcendental, while the second notion is stronger and so is called strongly transcendental or strictly transcendental.

Examples

• If $K$ is the rational numbers $\mathbb{Q}$ and $F$ is the complex numbers $\mathbb{C}$, then the transcendental elements in $\mathbb{C}$ are precisely the transcendental numbers.

• If $K$ is the rational numbers $\mathbb{Q}$ and $F$ is the p-adic complex numbers $\mathbb{C}_p$, then the transcendental elements in $\mathbb{C}_p$ are precisely the transcendental p-adic numbers.

Related concepts

• algebraic element

• transcendental number

• transcendental extension

• algebraically independent subset

References

See also

• Wikipedia, Transcendental element