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Scherie
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Let E be an extension of F and let a, b belong to E. Prove that F(a,b) =F(a)(b) = F(b)(a).
Deveno said:May we assume that $a$ and $b$ are algebraic over $F$?
In field extensions, F(a,b) is defined as the smallest field that contains both a and b as elements. It is denoted as F(a,b) = F(a)(b) = F(b)(a).
Proving F(a,b)=F(a)(b)=F(b)(a) in field extensions is significant because it shows that the field extension is commutative, meaning that the order in which the elements are multiplied does not affect the result.
The proof for F(a,b)=F(a)(b)=F(b)(a) in field extensions involves showing that both sides of the equation have the same elements and satisfy the same properties, such as closure, associativity, and commutativity of addition and multiplication.
One example of F(a,b)=F(a)(b)=F(b)(a) in field extensions is the field extension Q(sqrt(2), sqrt(3)), where Q represents the rational numbers. In this case, F(a,b)=F(a)(b)=F(b)(a)=Q(sqrt(2), sqrt(3)).
Proving F(a,b)=F(a)(b)=F(b)(a) in field extensions plays a crucial role in Galois theory, which studies the structure of field extensions. It helps to understand the behavior of roots of polynomials and the solvability of equations, among other applications.