Math 2.0

Field theory and polynomials Theorem

The reals form a field

Builds on Real numbers

Statement

The real numbers ℝ\mathbb{R}, with their addition and multiplication, form a field. That is, ℝ\mathbb{R} is a commutative ring with 0≠10 \neq 1, and each real number x≠0x \neq 0 has a multiplicative inverse x−1x^{-1} with x⋅x−1=1x \cdot x^{-1} = 1.

Remarks

The field ℝ\mathbb{R} contains the field ℚ\mathbb{Q} of rational numbers as a subfield. In the axiomatic approach, this is part of the axioms. In a construction of ℝ\mathbb{R}, for example with Cauchy sequences, it is a theorem.

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Where this leads

21 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Real_number, Arithmetic
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Tag0FGENPMPJX
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Kindstatement, theorem
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MSC12D99
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UsesReal numbers

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