Algebraic conjugation

Elements of a field are called algebraically conjugated if they have the same minimal polynomial with respect to a subfield .

definition

Let be a field extension and the polynomial ring to with the indefinite . Let the elements be algebraically about , that is, they exist with . ${\displaystyle L/K}$ ${\displaystyle K[x]}$ ${\displaystyle K}$ ${\displaystyle x}$ ${\displaystyle a,b\in L}$ ${\displaystyle K}$ ${\displaystyle 0\neq q(x),p(x)\in K[x]}$ ${\displaystyle p(a)=q(b)=0}$

Then and are algebraically conjugated over if and have the same minimal polynomial over . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle K}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle K}$

If the connection is clear, the term “conjugated” is also used for a shorter time.

properties

• ${\displaystyle a}$and are conjugated over the body if and only if that applies to all . ${\displaystyle b}$ ${\displaystyle K}$ ${\displaystyle p(x)\in K[x]}$ ${\displaystyle p(a)=0\Leftrightarrow p(b)=0}$
• Let be a finite body extension with for one . Then are conjugated over the field if and only if there is an element in the Galois group with . ${\displaystyle L/K}$ ${\displaystyle L=K(b)}$ ${\displaystyle b\in L\setminus K}$ ${\displaystyle a,b\in L}$ ${\displaystyle K}$ ${\displaystyle \varphi }$ ${\displaystyle \mathrm {Gal} (L/K)}$ ${\displaystyle \varphi (a)=b}$

Examples

• The complex numbers and have the minimal polynomial over both and are therefore algebraic conjugate over . About course they have the minimal polynomials and and are not conjugated. ${\displaystyle i}$ ${\displaystyle -i}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \textstyle x^{2}+1}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle x-i}$ ${\displaystyle x+i}$
• More generally, the following applies: Two complex numbers and with are algebraically conjugated over if and only if they emerge from each other through complex conjugation , so the following applies. The common minimal polynomial in this case is . ${\displaystyle a+bi}$ ${\displaystyle c+di}$ ${\displaystyle b,d\neq 0}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle a=c,b=-d}$ ${\displaystyle x^{2}-2ax+a^{2}+b^{2}}$
• The golden number and its negative reciprocal are conjugated over the body . They are solutions of the minimal polynomial . ${\displaystyle \Phi }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \textstyle x^{2}-x-1}$
• The algebraic conjugates are obtained as follows: From ${\displaystyle x_{1}={\sqrt {2}}+{\sqrt {3}}}$

${\displaystyle x_{1}^{2}=5+2{\sqrt {6}}}$, and ${\displaystyle \quad x_{1}^{3}=11{\sqrt {2}}+9{\sqrt {3}}\quad }$ ${\displaystyle \quad x_{1}^{4}=49+20{\sqrt {6}}}$

the minimal polynomial results

${\displaystyle x^{4}-10x^{2}+1=\left(x^{2}-5\right)^{2}-24}$.

By pulling the roots twice, together with the relationship , the other zeros are obtained: ${\displaystyle 5\pm 2{\sqrt {6}}=({\sqrt {2}}\pm {\sqrt {3}})^{2}}$

${\displaystyle x_{2}={\sqrt {2}}-{\sqrt {3}}}$, , . ${\displaystyle \quad x_{3}=-{\sqrt {2}}+{\sqrt {3}}}$ ${\displaystyle \quad x_{4}=-{\sqrt {2}}-{\sqrt {3}}}$