Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [totally-real-field]

For questions about number fields that are totally real, i.e. whose every complex embedding has real image.

A number field is totally real if the image every complex embedding is contained in the reals. Equivalently, if it is generated over the rationals by one root of a polynomial with only real roots. Totally real fields are closely related to CM-fields, which are (non-real) quadratic extensions of totally real fields.

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Finding an irreducible polynomial of degree $n$ in $\mathbb Q[X]$ with real roots

Context. I was trying to prove that for a given $n$, there exist a totally real number field of degree $n$. I understood that it was equivalent to find a polynomial $P$ such that (i) $P\in\mathbb Q[X]$ ; (ii) $P$ is irreducible ; (iii) $P$ only has…
E. Joseph
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Nonstandard analysis and hyperreals

Hyperreal numbers and nonstandard analysis are intimately connected. First, the nonstandard analysis consists of an extension of the axioms for the real numbers, which introduces a new predicate $\text{st}(x)$ (read: "$x$ is a standard real…
Davius
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Example of a totally real number that is neither totally positive nor totally negative?

Say that an algebraic number $\alpha\in \mathbb{C}$ is totally real iff $\mathbb{Q}(\alpha)$ is a totally real number field. Why does the set of all totally real numbers form a subfield of $\mathbb{R}$? What is an example of a totally real number…
mathematick
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Does there getting from $1$ to $\sqrt[4]{2}$ using $\sqrt{\alpha^2+ 1}$

The aim is to get from $1$ to $\sqrt[4]{2}$ or prove it is impossible using only one of the following options: Add or subtract two previously constructed numbers. Multiply two previously constructed numbers. Using a previously constructed number…
razivo
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Ramification in Maximal Totally Real Subfield

Exercise 12, Chapter 4 of Marcus' Number Fields asks the following: Let $p$ be a prime not dividing $m$. Determine how $p$ splits in $\mathbb{Q}(\zeta_m +\zeta_m^{-1})$. Certainly $p$ is unramified, but I'm having trouble finding what the inertial…
a196884
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