In field theory, particularly in the context of abstract algebra and number theory, the concept of a "conjugate element" often refers to the behavior of roots of polynomials and their extensions in fields. ### Conjugate Elements in Field Theory 1. **Field Extensions**: When we have a field extension \( K \subset L \), elements of \( L \) that are roots of a polynomial with coefficients in \( K \) are called conjugates of each other.
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The algebraic conjugates of an algebraic element over a field are the polynomial roots of its algebraic minimal polynomial over in an algebraic closure.
The algebraic conjugates of an algebraic number are the polynomial roots of its algebraic minimal polynomial, equivalently its images under field embeddings of its number field into an algebraic closure.