General linear group over subfield is conjugacy-closed

Statement
Let $$k$$ be a subfield of a field $$K$$. Then $$GL_n(k)$$ (the general linear group, the group of invertible $$n \times n$$ matrices with entries in $$k$$) is a subgroup of $$GL_n(K)$$ (the group of invertible $$n \times n$$ matrices with entries in $$K$$). This subgroup is conjugacy-closed: in other words, if two elements of $$Gl_n(k)$$ are conjugate in $$GL_n(K)$$, they are also conjugate in $$GL_n(k)$$.

Related facts about general linear group

 * General linear group of subspace is conjugacy-closed
 * GL IAPS is concatenation-conjugacy-closed
 * Orthogonal group is conjugacy-closed in general linear group over reals
 * Unitary group is conjugacy-closed in general linear group over complex numbers
 * Brauer's permutation lemma

Opposite facts about general linear group

 * General linear group of subring of field need not be conjugacy-closed