The reals form a field

Depends on

• The real numbers
• Cauchy sequences form a commutative ring
• Null sequences form an ideal
• The null ideal is maximal
• Null sequence

Used by

• For every fixed finite size at least one, the determinant of a real square matrix is a polynomial in its matrix entries Corollary
• The first quadrant of ℝ² contains 0 and is closed under addition and is not a linear subspace, since it is not closed under multiplication by -1 Counterexample
• For n≥1, the determinant over a commutative ring by the Leibniz formula, and |det A| for a real matrix Definition
• The quaternions ℍ: real quadruples with componentwise addition and an explicit multiplication formula matching the table on 1, i, j, k Definition
• A consistent underdetermined system has an affine two-parameter solution set Example
• A nonidentity projection of the plane has determinant zero and is not invertible Example
• For any field F, (F, +) and (F ∖ {0}, ·) are abelian groups; in particular (ℚ, +), (ℚ ∖ {0}, ·), (ℝ, +) and (ℝ ∖ {0}, ·) Example
• One operator has matrices diag(2,3) and beginpmatrix2&01&3 endpmatrix in two bases, both with determinant 6 Example
• ℚ and ℝ are fields, hence commutative rings, integral domains and ordered rings, all of characteristic 0 Example
• ℝ as a vector space over ℚ has a basis, and every such basis is infinite; the existence proof exhibits none Example
• ℝ is a vector space over itself, over the embedded copy of ℚ by restriction of scalars, and over ℚ itself via the embedding Example
• The quarter-turn (x,y)↦(-y,x) on ℝ² has matrix beginpmatrix0&-11&0 endpmatrix and square -I₂ Example
• The rank and solution behaviour of a parameterised matrix change at one exceptional parameter Example
• The reals are the quotient of rational Cauchy sequences by the maximal ideal of null sequences Example
• The vector (1,2) ∈ ℝ² has coordinate list (1,2) in the standard ordered basis, (2,1) in its reversal, and (2,-1) in the ordered basis ((1,1),(1,0)) Example
• FALSE: det(A+B)=det(A)+det(B) for all same-sized square matrices False statement
• The rationals embed densely in the reals Lemma
• A finite square real matrix is invertible if and only if its determinant is nonzero Theorem
• Brownian scaling Theorem
• Every invertible finite square real matrix is a finite product of elementary matrices Theorem
• ℍ is a division ring that is not commutative, hence not a field: q⁻¹ = q̄ / N(q) for q ≠ 0, while ij = k and ji = -k Theorem
• The reals form a totally ordered field Theorem

Cited to discharge well-definedness by The real numbers.

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• T. Tao, Analysis I, 3rd ed., §5.3 (standard reference, not scraped)
• W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
• L. S. Krapp, Constructions of the real numbers: a set theoretical approach (Oxford, 2014) (standard reference, not scraped)