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Palindrom
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Suppose k is separable over E. Prove that E(k)/E is separable.
A hint, if I may ask.
A hint, if I may ask.
Separability in this context refers to the property of a field extension (E(k)/E) where every element in the extension can be obtained by adjoining a root of a separable polynomial over the base field E.
Proving separability is important because it allows us to determine whether a field extension is normal, which has implications for algebraic closure and the existence of Galois groups.
A polynomial is separable if its roots are distinct in the algebraic closure of the base field. In the context of E(k)/E, separability of the extension is equivalent to the existence of a separable polynomial with roots in the extension.
One example of a separable field extension is the extension of the rational numbers by adjoining the square root of 2 (i.e. Q(√2)/Q). This extension is separable because the minimal polynomial of √2 over Q, x^2 - 2, is a separable polynomial.
One common technique is to use the fact that every irreducible polynomial has a separable factor, which allows us to decompose a polynomial into separable factors. Another approach is to use the Frobenius map to show that the extension is separable.