definition. affine space [gathmann2013commutative, 0.3] [ag-000A]

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The (\(n\)-dimensional) affine space over a field \(K\), denoted \(\mathbb {A}_K^n\), is \[ \left \{\left (c_1, \ldots , c_n\right ): c_i \in K \text { for } i=1, \ldots , n\right \} \] which is just \(K^n\) as a set, without the its additional structures as a \(K\)-vector space and a ring.

We'll often use the term affine \(n\)-space to indicate the dimension. Particularly, an affine 1-space is called an affine line, an affine 2-space is an affine plane.