Suppose that E is an extension of F and a, b ∈E. If a is algebraic over F of degree m, and b is

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Suppose that E is an extension of F and a, b ∈E. If a is...

Suppose that $E$ is an extension of $F$ and $a, b \in E$. If $a$ is algebraic over $F$ of degree $m$, and $b$ is algebraic over $F$ of degree $n$, where $m$ and $n$ are relatively prime, show that $[F(a, b): F]=m n$.

Key Concepts

Coprime Degrees and Linear Disjointness

When two algebraic extensions of a field F have degrees that are relatively prime, they are often linearly disjoint. This means that the only common subfield they share is F itself, which results in the degree of the composite extension being the product of the individual degrees. This property simplifies the computation of the degree of the composite field and is key in proving results like [F(a, b) : F] = m*n.

Tower Law

The Tower Law, or the multiplicativity of degrees, states that if F ? K ? E are fields, then the degree of the extension E over F is the product of the degrees of E over K and K over F. This law is a crucial tool in computing and relating the degrees of composite field extensions.

Field Extensions

A field extension is a pair of fields E and F such that F is a subfield of E. Field extensions allow us to study the properties of a larger field in relation to a smaller one, and are fundamental in understanding algebraic structures, including how new elements are introduced and how they interact with the base field.

Degree of Algebraic Extensions

The degree of an algebraic extension is the dimension of the larger field as a vector space over the smaller field. When an element is algebraic over a field F, its minimal polynomial has a finite degree, and this degree is exactly the degree of the field extension generated by that element. This concept is essential in determining how ‘big’ an extension is compared to its base field.

Minimal Polynomials

The minimal polynomial of an algebraic element over a field F is the monic polynomial of least degree with coefficients in F that the element satisfies. It is irreducible over F, and its degree gives the degree of the element’s field extension. Minimal polynomials are central in understanding the algebraic dependencies in field extensions.

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