Every element of a finite field is expressible as a sum of two squares
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Contents
History
This result is attributed to Henry Mann.
Statement
Suppose is a finite field. Then, every element
can be expressed in the form
, where
.
Facts used
- Multiplicative group of a finite field is cyclic: We actually only need the weaker statement that, for a field of odd characteristic, exactly half the elements of the multiplicative group are squares.
- Product of subsets whose total size exceeds size of group equals whole group: If
are subsets of a finite group
, where
, then
.
Proof
Case of characteristic two
In this case, the square map is surjective and every element is a square, because the multiplicative group is of odd order.
Case of odd characteristic
- Reasoning in the multiplicative group: Suppose
has
elements. Then its multiplicative group
has
elements. By fact (1), the multiplicative group is cyclic of order
, which is even. Thus, exactly half the elements (corresponding to even powers of the generator) are squares. Since
is also a square, we obtain
elements of
that are squares.
- Reasoning in the additive group: We now apply fact (2) with
as the additive group of
, which has size
, and both
and
as equal to the set of (multiplicative) squares, which has size
.