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.

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

1 Answer 1


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 $\mathbb{R}$).

  • $\begingroup$ A field is also a vector space over the prime field and any intermediary filed. $\endgroup$
    – lhf
    Jul 31, 2014 at 20:40

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