What is an example of a number field with signature $(4,0)$?

Question

I know why, say, quadratic fields have signature $(2,0)$ or complex number fields have signature $(0,1)$. But what does a number field with signature $(4,0)$ look like?

Answer 1

If a number field is given by a defining polynomial as $K = \mathbf Q[x]/(f(x))$ then the number of real and complex embeddings are just the number of real and complex roots of this polynomial respectively. So to give a number field with 4 real embeddings you can just give a monic irreducible polynomial with rational coefficients with 4 real roots and no complex ones, so a degree 4 polynomial with only real roots, you can find one just by playing with the coefficients and looking at a plot I'm sure!

Some nice less random examples would be to take biquadratic extensions, by adjoining square roots of a pair of different enough positive integers, i.e.

$$\mathbf Q (\sqrt{3}, \sqrt{5})$$

gives a degree 4 extension and no matter how you embed the generators into $\mathbf C$ the image is always contained in the reals.

Alternatively you could take a totally positive element of a real quadratic field (this is one for which both conjugates are positive) and adjoin a square root of that. For example inside $\mathbf Q (\sqrt{13})$ the element $4 + \sqrt{13}$ is totally positive as under whichever embedding we choose both $4+ \sqrt{13}$ and $4 - \sqrt{13}$ are positive (because $4^2 = 16 > 13$ so the positive real value of $\sqrt{13} < 4$). So we can add a square root of $4 + \sqrt{13}$ to get $$\mathbf Q(\sqrt{4 + \sqrt{13}})$$ a degree 4 totally real extension (totally real is the name for the fields you are interested in of degree $n$ with signature $(n,0)$.)

As the minimal polynomial of $4+\sqrt{13}$ is $x^2 - 8x + 3$ a defining polynomial for this field is $x^4 - 8x^2 + 3$.

If you want to see a lot more examples of such fields you can search for some on the LMFDB like this http://www.lmfdb.org/NumberField/?signature=%5B4%2C0%5D though it might be hard to find a "nice" description of some of these.

Answer 2

You can for example take $\mathbb{Q}(\sqrt{p},\sqrt{q})$ for some distinct primes $p,q$. Then you will have $4$ real embeddings via the four combinations of $\sqrt{p} \mapsto \pm \sqrt{p}$ and $\sqrt{q} \mapsto \pm \sqrt{q}$.