Articles of extension field

Field extensions and monomorphisms

I am working through some algebra exercises and got stuck with the following problem: I am working in the finite field $\mathbb{Z}_5$. Let $f \in \mathbb{Z}_5[x]$ and $f(x) = x^2 + 2$. By simple trial and error, I checked that this polynomial is irreducible, hence $\mathbb{Z}_5/(f) \cong GF(5^2)$, denoting the Galois field ($f$ is of […]

If $F(\alpha)=F(\beta)$, must $\alpha$ and $\beta$ have the same minimal polynomial?

Let’s consider a field $F$ and $\alpha,\beta\in\overline{F}-F$ (where $\overline{F}$ is an algebraic closure). If $F(\alpha)=F(\beta)$, is it true that $\alpha$, $\beta$ have the same minimal polynomial over $F$? For reference, I explain my way. (From here forward, all field isomorphisms are the identity on $F$.) There exist polynomials $f,g\in F[x]$ such that $f(\beta)=\alpha$, $g(\alpha)=\beta$. Let […]

Is the embedding problem with a cyclic kernel always solvable?

This question comes from this question by user72870. I shall explain how it relates to that question at the end. Let me shortly define my question: We call an embedding problem a diagram of the form: $$\begin{matrix}&&\mathfrak G\\&&\downarrow\\G&\rightarrow&\Gamma\end{matrix},$$ where $G\rightarrow\Gamma$ is a group extension (therefore we assume that the homomorphism is surjective), and $\mathfrak G$ […]

Computing the quotient $\mathbb{Q}_p/(x^2 + 1)$

This question already has an answer here: Decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}[i]$ into a product of fields 1 answer

Galois extension of $\mathbb{Q}$ with Galois group $\mathbb{Z}/4\mathbb{Z}$ that contains $\mathbb{Q}(\sqrt{3})$?

Does there exist a normal extension $L ⊃ \mathbb{Q}(\sqrt3) ⊃ \mathbb{Q}$ with Galois group $\mathrm{Gal}(L/\mathbb{Q}) \cong \mathbb{Z}/4\mathbb{Z}$?

Suppose $K/F$ is an algebraic extension of fields. Prove that if $R$ is a ring with $F ⊆ R ⊆ K$, then R must also be a field.

This question already has an answer here: Intermediate ring between a field and an algebraic extension. 3 answers

Generators of $\mathbb{Z}$ and $\mathbb{Z}$ when $\mathbb{Z}$, $\mathbb{Z}$ are f.g.

I want to show that $\mathbb{Z}[\alpha.\beta]$ and $\mathbb{Z}[\alpha\pm\beta]$ are finitely generated when $\mathbb{Z}[\alpha]$, $\mathbb{Z}[\beta]$ are f.g. My primary aim is to show the set of algebraic integers of a number field $K$ is a ring. We have that $\alpha,\beta$ are algebraic integers, so $\mathbb{Z}[\alpha]$ and $\mathbb{Z}[\beta]$ are finitely generated, so there exist minimal polynomials $f_1$,$f_2$ […]

A separable field extension of degree a product of two primes

The inspiration for asking this question is due to this question and its first answer; see also this same question: Let $F$ be a field and let $E/F$ be a separable field extension with $[E:F]=n=p_1p_2$, where $p_1$ and $p_2$ are primes (not necessarily different primes). Let $\alpha_1$ be a primitive element: $E=F(\alpha_1)$. Assume that $\alpha_1 […]

Field extensions and monomorphism

Suppose $[E_1:F]=m<\infty$ and $E_1$ is algebraic extension of $F$. If $K$ is any extension of $F$ then the number of monomorphism of $E_1/F$ into $K/F$ is at most $m$. I am trying to prove this by induction on $[E_1:F]$. If I pick an irreducible polynomial $g(x)$ with degree $m$ over $F$ and look at the […]

Writing elements of field extension in terms of the basis determined by a root of a polynomial

Let $\alpha \in \mathbb{C}$ be a root of the irreducible polynomial $$f(X) = X^3 + X + 3$$ Write the elements of $\mathbb {Q}(\alpha)$ in terms of the basis $\{1, \alpha, \alpha^2\}$. The first part is to work out $\alpha^3$ in terms of the basis, but I can’t work out if I need to explicitly […]