Intereting Posts

Are all non-squares generators modulo some prime $p$?
Expected number of steps to walk through points by multiple walkers
A question on morphisms of fields
Two homogenous system are equivalent if they have the same answer
understanding the basic definition
The number of words that can be made by permuting the letters of _MATHEMATICS_ is
Logic for decomposing permutation into transpositions
Difference between axioms, theorems, postulates, corollaries, and hypotheses
Dirac delta in polar coordinates
The product of a cofibration with an identity map is a cofibration
Do sets, whose power sets have the same cardinality, have the same cardinality?
Writing an expression as a sum of squares
A mouse leaping along the square tile
Michaelis-Menten steady state hypothesis
Intersection of sets of positive measure

What is(are) the Galois Extension(s) of $\mathbb{Q}$ whose Galois group is cyclic group of prime order? The fundamental theorem of Galois theory says that the degree of the extension is same as the order of the Galois group.Can we find an explicit polynomial of degree which is an arbitrary prime number?

- Prove that $\operatorname{Gal}(\mathbb{Q}(\sqrt{2}, i)/\mathbb{Q}(\sqrt{-2})) \cong Q_8$
- Monic irreducible polynomials of degree 6 in $F_{5}$
- What is a maximal abelian extension of a number field and what does its Galois group look like?
- Galois group of $X^4 + 4X^2 + 2$ over $\mathbb Q$.
- On Galois groups and $\int_{-\infty}^{\infty} \frac{x^2}{x^{10} - x^9 + 5x^8 - 2x^7 + 16x^6 - 7x^5 + 20x^4 + x^3 + 12x^2 - 3x + 1}\,dx$
- Conditions on cycle types for permutations to generate $S_n$
- Is it true that every element of $\mathbb{F}_p$ has an $n$-th root in $\mathbb{F}_{p^n}$?
- Cyclotomic polynomials and Galois groups
- Radical extension and discriminant of cubic
- Splitting field of a separable polynomial is separable

To construct an Galois extension of $\mathbb{Q}$ of order $p$, take an integer $N$ such that $(\mathbb{Z}/N\mathbb{Z})^*$ maps surjectively onto $\mathbb{Z}/p\mathbb{Z}$, then the field $\mathbb{Q}(\zeta_N)$ contains a subfield $K$ such that $[K:\mathbb{Q}]=p$ and $K$ is Galois over $\mathbb{Q}$ (Here $\zeta_N$ is a primitive $N$-th root of unity). For example, to find a galois extension of $\mathbb{Q}$ of degree 5, you may look at the subfield of $\mathbb{Q}(\zeta_{11})$ that’s fixed by the complex conjugation. Kronecker-weber Theorem states that every finite abelian extensions of $\mathbb{Q}$ is a subfield of cyclotomic fields, so every Galois extension of $\mathbb{Q}$ of prime order arises this way.

Continue the example with prime $p=5$, to find all of such extensions, we will look at the sequence $5n+1$. By Dirichlet’s theorem on primes in arithmetic progressions, there are infinitely many primes $q$ in this arithmetic progression, each $\mathbb{Q}(\zeta_q)$ contains a subfield which is a degree 5 extension of $\mathbb{Q}$. All these subfields are distinct because they ramifies at different primes. Last, note that that there is also a subfield of $\mathbb{Q}(\zeta_{25})$ which is of degree 5 over $\mathbb{Q}$.

Edit: I was wrong about my complete list statement. For example, $\mathbb{Q}(\zeta_{341})$ (composite of $\mathbb{Q}(\zeta_{11})$ and $\mathbb{Q}(\zeta_{31})$) has 6 subfields which are degree 5 abelian extensions of $\mathbb{Q}$. Apologies.

- Bellard's exotic formula for $\pi$
- Can a field be isomorphic to its subfield but not to a subfield in between?
- Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$
- Prove that $a$ is quadratic residue mod $m$ iff $a$ is quadratic residue mod $p$ for each prime $p$ divides $m$
- Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
- Uncountable ordinals without power set axiom
- Showing non-cyclic group with $p^2$ elements is Abelian
- Can all real/complex vector spaces be equipped with a Hilbert space structure?
- Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$
- Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$
- When does it help to write a function as $f(x) = \sup_\alpha \phi_{\alpha}(x)$ (an upper envelope)?
- Proving $\sum\limits_{i=0}^n i 2^{i-1} = (n+1) 2^n – 1$ by induction
- Proving two measures of Borel sigma-algebra are equal
- Is $\mathbb{Z}/p^\mathbb{N} \mathbb{Z}$ widely studied, does it have an accepted name/notation, and where can I learn more about it?
- What is reductive group intuitively?