Describe the smallest subfield of the field of real numbers...

Describe the smallest subfield of the field of real numbers that contains $\sqrt{2}$. (That is, describe the subfield $K$ with the property that $K$ contains $\sqrt{2}$ and if $F$ is any subfield containing $\sqrt{2}$, then $F$ contains $K .$ )

Key Concepts

Field

A field is an algebraic structure consisting of a set equipped with two binary operations: addition and multiplication. These operations satisfy properties such as associativity, commutativity, the existence of identity and inverse elements, and the distributive property. This concept is central to understanding many areas of algebra, including the structure of number systems like the rational and real numbers.

Subfield

A subfield is a subset of a field that is itself a field under the same operations. To be a subfield, the subset must contain the additive and multiplicative identities and be closed under addition, subtraction, multiplication, and division (by nonzero elements). Subfields are important in analyzing and constructing smaller or simpler fields within larger fields.

Field Extension

A field extension is a larger field constructed from a smaller field by adjoining additional elements that are not originally in the smaller field. It involves creating a new field that contains both the elements of the base field and additional elements that satisfy certain algebraic properties. Field extensions allow mathematicians to explore solutions to equations that might not exist within the original field.

Algebraic Element

An algebraic element over a field is an element that satisfies a nonzero polynomial equation with coefficients in that field. In the context of field extensions, adjoining an algebraic element to a field yields a field where the element and its algebraic properties are fully incorporated. This is a key idea when constructing the smallest subfield that contains a given algebraic element, as it leads to a field expressed in terms of polynomial expressions with rational coefficients.

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