The basis of extension field contains $1$

Question

Let $E$ be a extension of field $F$. It is known that $E$ may be considered as vector space over $F$. Is it always possible to find a basis in $E$ that contains $1$? Maybe it is possible when $E$ is finite extension?

Answer 1

For any vector space $V$, if $\{x_1, \dots, x_n\}$ is a linearly independent set, then there exists elements $x_{n+1}, \dots, x_m$ (with possibly $m = n$) such that $\{x_1, \dots, x_m\}$ forms a basis.

Since $\{1\}$ is a linearly independent set, the statement follows.