# Complex conjugate and Field Extension

Let $$E$$ be a subfield of $$\mathbb{C}$$ and Let $$\overline{E}=\{\overline{z} \, |\, z \in E \}$$ with $$\overline{z}$$ being the complex conjugate of $$z$$. Let $$K$$ be a subfield of $$\mathbb{C}$$ with $$\overline{K}=K$$ and $$w\in \mathbb{C}$$ and $$w^2 \in K$$. Is (when is) $$\overline{K(w)}=K(w)$$?

I have no idea how to show it and would be thankful for hints (please no solutions at this point). Some things I know: Let $$w\notin K$$. Since $$w\in \mathbb{C}$$ and $$w^2 \in K \Rightarrow [K(w):K]=2 \Rightarrow$$ Minimalpolynomial of $$w$$ over $$K$$ has degree $$2$$. I also know that $$\mathbb{C}$$ is algebraic closed.

## 2 Answers

Hint(s): You know that $$[K(\omega):K]=2$$- what does that mean for what the elements of $$K(\omega)$$ will look like?- also, notice that the function that takes conjugates has some nice algebraic properties. Finally, if $$\omega$$ is a root of a quadratic over $$K$$, say, $$x^2+ax+b$$ (with $$a$$ and $$b\in K$$) what can you say about $$\overline\omega$$?
(extra hint below)

What field can you find $$\overline\omega$$ in?- look at the specific fact that $$\omega^2\in K$$ and find a condition on $$\omega$$'s components

First of all thank you for taking the time and helping:

1.) $$[K(w):K]=2 \Rightarrow \{1,w\}$$ is $$K$$-basis of $$K(w)$$. Every element of $$K(w)$$ can be written as $$a+bw$$ with $$a,b\in K$$.

2.) The function that takes conjugates is a homomorphism of field. It's easy to verify that $$\overline{a+b}=\overline{a}+\overline{b}$$ and $$\overline{a\cdot b}=\overline{a}\cdot \overline{b}$$. This might also mean that the function implicates an extension from $$K(w)\rightarrow \mathbb{C}$$ because $$\overline{K}=K$$?

3.) If $$w$$ is a root of that polynomial, isn't $$\overline{w}$$ also a root since because $$a,b \in K$$ and $$\overline{K}=K$$?

From here on I just don't know what to do...

• For item 3: Consider the case $K=\Bbb{Q}(i)$. Let $w=\sqrt{2+i}$ (to be specific, let's use the square root in the first quadrant). Then $w^2\in K$. Furthermore, $w$ is a zero of the polynomial $x^2-(2+i)$ (which has coefficients in $K$). But $\overline{w}$ is not a zero of that polynomial. Instead, it is a zero of the polynomial $x^2-(2-i)$. – Jyrki Lahtonen Jan 17 at 6:43