Editing Minkowski's theorem (section)

===Applications to number theory===
====Primes that are sums of two squares====
The difficult implication in [[Fermat's theorem on sums of two squares]] can be proven using Minkowski's bound on the shortest vector.

'''Theorem:''' Every [[prime number|prime]] with <math display="inline">p \equiv 1 \mod 4</math> can be written as a sum of two [[square number|squares]].

{{math proof | proof = Since <math display="inline">4 \,|\, p - 1</math> and <math display="infline">a</math> is a [[quadratic residue]] modulo a prime <math display="inline">p</math> if and only if <math display="infline"> a^{\frac{p-1}{2}} = 1~(\text{mod}~p)</math> (Euler's Criterion) there is a square root of <math display="inline">-1</math> in <math display="inline">\Z / p\Z</math>; choose one and call one representative in <math display="inline">\Z</math> for it <math display="inline">j</math>. Consider the lattice <math display="inline">L</math> defined by the vectors <math display="inline">(1, j), (0,p)</math>, and let <math display="inline">B</math> denote the associated [[matrix (mathematics)|matrix]]. The determinant of this lattice is <math display="inline">p</math>, whence Minkowski's bound tells us that there is a nonzero <math display="inline">x = (x_1, x_2) \in \mathbb{Z}^2</math> with <math display="inline">0 < \|Bx\|_2^2 < 2p</math>. We have <math display="inline">\|Bx\|^2 = \|(x_1, jx_1 + px_2)\|^2 = x_1^2 + (jx_1 + px_2)^2</math> and we define the integers <math display="inline">a = x_1, b = (jx_1 + px_2)</math>. Minkowski's bound tells us that <math display="inline">0 < a^2 + b^2 < 2p</math>, and simple [[modular arithmetic]] shows that <math display="inline">a^2 + b^2 = x_1^2 + (jx_1 + px_2)^2 = 0 \mod p</math>, and thus we conclude that <math display="inline">a^2 + b^2 = p</math>. Q.E.D.}}

Additionally, the lattice perspective gives a computationally efficient approach to Fermat's theorem on sums of squares:
{{Collapse top|title= Algorithm}} First, recall that finding any nonzero vector with norm less than <math display="inline">2p</math> in <math display="inline">L</math>, the lattice of the proof, gives a decomposition of <math display="inline">p</math> as a sum of two squares. Such vectors can be found efficiently, for instance using [[Lenstra–Lenstra–Lovász lattice basis reduction algorithm|LLL-algorithm]]. In particular, if <math display="inline">b_1, b_2</math> is a <math display="inline"> 3/4 </math>-LLL reduced basis, then, by the property that <math display="inline">\|b_1\| \leq (\frac{1}{\delta - .25})^{\frac{n-1}{4}} \text{det}(B)^{1/n}</math>, <math display="inline">\|b_1\|^2 \leq \sqrt{2} p < 2p</math>. Thus, by running the LLL-lattice basis reduction algorithm with <math display="inline"> \delta = 3/4 </math>, we obtain a decomposition of <math display="inline">p</math> as a sum of squares. Note that because every vector in <math display="inline">L</math> has norm squared a multiple of <math display="inline">p</math>, the vector returned by the LLL-algorithm in this case is in fact a shortest vector.
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====Lagrange's four-square theorem====
Minkowski's theorem is also useful to prove [[Lagrange's four-square theorem]], which states that every [[natural number]] can be written as the sum of the squares of four natural numbers.

====Dirichlet's theorem on simultaneous rational approximation====

Minkowski's theorem can be used to prove [[Dirichlet's approximation theorem|Dirichlet's theorem on simultaneous rational approximation]].

====Algebraic number theory====
Another application of Minkowski's theorem is the result that every class in the [[ideal class group]] of a [[number field]] {{math|''K''}} contains an [[integral ideal]] of [[field norm|norm]] not exceeding a certain bound, depending on {{math|''K''}}, called [[Minkowski's bound]]: the finiteness of the [[Class number (number theory)|class number]] of an algebraic number field follows immediately.