# What is an algebraic extension of a field F?

Definition. An extension field E of a field F is an algebraic extension of F if every element in E is algebraic over F. Definition. If an extension field E of a field F is of finite dimension n as a vector space over F, then E is a finite extension of degree n over F. We let [E : F] denote the degree of E over F.

## Is R algebraic extension of Q?

Example. Q(√2) and Q(√3) are algebraic extensions of Q. R is not an algebraic extension of Q.

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## Is an algebraic extension a finite extension?

A finite extension is algebraic. In fact, an extension E/k is algebraic if and only if every subextension k(\\alpha )/k generated by some \\alpha \\in E is finite. In general, it is very false that an algebraic extension is finite.

## Is a field extension a field?

For this purpose, one abstractly defines a field extension as an injective ring homomorphism between two fields. Every non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper ideals, so field extensions are precisely the morphisms in the category of fields.

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## [KEY]Is C algebraic over Q?[/KEY]

2. False. C is not an algebraic extension of Q, so by definition of algebraic closure it cannot be an algebraic closure of Q. The fact that this is a transcendental extension can be stated by proving, for instance, that e or π are not algebraic.

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## Is C algebraic extension of R?

Any algebraic extension of R is contained in C, as C is the algebraic closure of R. Therefore, C is the only extension of R of degree 2, up to R-isomorphisms.

## Is Q an extension field of Z2?

T F “Q is an extension field of Z2.” False: Z2 is not a subfield of Q because its operations are not induced by those of Q. (Moreover, Z2 cannot even be isomorphic to a subfield of Q because the char- acteristics are different.)