# Are all fields vector spaces?

1. Are $\mathbb{Z_p},\mathbb{Q},\mathbb{R},\mathbb{C}$ above themselves vector space?

2. Is a field above anoother field a vector space?

As for 1. we know that $\Bbb R^n$ is a vector space so in particular it is true for $n=1$

For 2. by definition a vector space is $V$ with addition and multiplication over a field, therefore 2 is true.

• Obviously a field is a vector space over itself, or any subfield. Easy to verify.
– anon
Jul 31, 2014 at 19:54
• All fields are trivially vector spaces over themselves of dimension $1$. However, no proper sub-field is a subspace over the same field. Jul 31, 2014 at 19:54

A vector space is a set with an addition law and a scalar multiplication law, where the scalars are elements of a field. Thus, a vector space over a field may not be itself a field (e.g. continuous functions on an interval); however, a field is always a vector space over itself. Similarly, taking direct sums of a field will give you a vector space over the original field (e.g $\mathbb{R}^n$ is a vector space over the field $\mathbb{R}$).