Field theory and polynomials

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• ConceptField

  A field is a commutative ring in which you can divide by every nonzero element. The rational numbers, the real numbers and the complex numbers are fields. The definition of a field gives the axioms.

• ConceptSubfield

  A subfield of a field is a subset that is a field with the same addition and multiplication. The definition of a subfield lists the closure conditions.

• ConceptDivision ring

  A division ring is a nonzero ring in which every nonzero element has a multiplicative inverse. It is like a field, but the multiplication need not be commutative. The definition of a division ring gives the conditions.

• ConceptCharacteristic of a field

  The characteristic of a field is the least positive number of copies of $1$ that add up to $0$. It is $0$ when no such number exists. It is the characteristic of the field as a ring, and for a field it is always $0$ or a prime number. It decides the prime subfield, and many results of field theory depend on it. For example, the map $a \mapsto a^p$ is a ring homomorphism in characteristic $p$.

• ConceptPrime subfield

  The prime subfield of a field is its smallest subfield. It is $\mathbb{Q}$ or $\mathbb{F}_p$, up to isomorphism, by the characteristic. The definition of the prime subfield gives the details.

• ConceptField of fractions

  The field of fractions of an integral domain is the smallest field that contains it. Its elements are the fractions $a/b$ with $b \neq 0$. The rational numbers are the field of fractions of the integers. The definition of the field of fractions gives the construction.