Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that ...

learn more… | top users | synonyms (1)

7
votes
2answers
470 views

Determining subfields of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ without Galois theory?

Somehow I had convinced myself recently that one could determine the subfields of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (here $\zeta_3 = \frac{-1+\sqrt{3}i}{2}$ is a primtive $3$rd root of $1$) without ...
7
votes
1answer
337 views

For field extensions $F\subsetneq K \subset F(x)$, $x$ is algebraic over $K$

Let $x$ be an element not algebraic over $F$, and $K \subset F(x)$ a subfield that differs from $F$. Why is $x$ algebraic over $K$? Thanks a lot!
7
votes
1answer
689 views

Need help understanding separable polynomials

I have the following definition of what it means for a polynomial to be separable: Let $E$ be a field and $P \in E[x]$ be irreducible. Then we say $P$ is separable if it has no repeated root in ...
7
votes
1answer
95 views

Short method to prove irreduicibility of $x^7+21x^5+35x^2+34x-8$ over $\Bbb Q$?

I am given a task to prove that polynomial $f=x^7+21x^5+35x^2+34x-8$ is irreducible over $\Bbb Q$? In my algebra course we learnt reduction and Eisenstein criterion. Eisenstein doesn't seem to work ...
7
votes
2answers
455 views

Elementary proof of $\mathbb{Q}(\zeta_n)\cap \mathbb{Q}(\zeta_m)=\mathbb{Q}$ when $\gcd(n,m)=1$.

In an answer to another question I used the fact that $\mathbb{Q}(\zeta_m)\subseteq \mathbb{Q}(\zeta_n)$ if and only if $m$ divides $n$ (here $\zeta_n$ stands for a primitive $n$th root of unity, ...
7
votes
2answers
836 views

A question regarding normal field extensions and Galois groups

The following is possibly true but I can't find a corresponding theorem: If $E/F$ is the splitting field of some polynomial in $F$ and $F \subset K \subset E$ then: $Gal(E/K)$ normal subgroup of ...
7
votes
1answer
121 views

Infinite $p$-extension contains $\mathbb{Z}_p$-extension

Does the Galois group of every infinite $p$-extension $K$ of a number field $k$ contain a (closed) subgroup such that the quotient group is isomorphic to $\mathbb{Z}_p$? My feeling is "yes", but I'm ...
7
votes
1answer
130 views

Extension degree of residue field.

Let $k$ be a field, and $A$ be a finitely generated $k$-algebra with $\text{dim}(A)\leq 1$. Then for any maximal ideal $\mathfrak{m}$ of $A$, does this inequality $[A/\mathfrak{m}:k]<\infty$ hold? ...
7
votes
2answers
477 views

Derivations in a ring. What applications do they have outside algebra?

INTRODUCTION Let $R$ be any ring, possibly without unity. We call a function $d:R\longrightarrow R$ a derivation on $R$ if it satisfies the following conditions. $(1)$ It is an endomorphism of the ...
7
votes
1answer
218 views

Field Extensions

Let $L/K$ a finite extension and $f(x)\in K[x]$ a non-linear irreducible polynomial. Prove that if $\mathrm{gcd}\left( \mathrm{deg}(f) , \left[ L:K \right] \right)=1$ then $f(x)$ has no roots in ...
7
votes
3answers
306 views

Why are the surreals considered “recreational” mathematics?

One of my college math professors once remarked to me that it was interesting that John Conway's two "biggest" contributions to math were both recreational: the Game of Life and the Surreals. No one, ...
7
votes
3answers
186 views

Proving that $X^6-15X^4-6X^3+75X^2-90X-116$ is irreducible over $\mathbb Q$

When asked to find the minimal polynomial of $\sqrt[3]{3}+\sqrt[2]{5}$ over $ \mathbb Q$, I easily found out that $X^6-15X^4-6X^3+75X^2-90X-116$ has $\sqrt[3]{3}+\sqrt[2]{5}$ as a root. It's very ...
7
votes
1answer
137 views

A first order theory whose finite models are exactly the $\Bbb F_p$

Is there a first order theory $T$ in the language of rings such that its finite models are exactly the fields $\Bbb F_p$ with $p$ prime (but no $\Bbb F_q$ with $q$ a proper prime power is a model of ...
7
votes
2answers
125 views

Equivalent definition of algebraically closed

In Hungerford's Algebra text, it is stated that a field $K$ is algebraically closed iff there exists a subfield $F$ such that $K$ is algebraic over $F$ and all polynomials in $F[x]$ split in $K[x]$. ...
7
votes
1answer
140 views

prove that the field extension is cyclic

Let's define the sequence $x_0=0$ and $ x_{i+1} = \sqrt{x_i+2}$ taking always the positive root. Prove that the field extension $\Bbb Q \subset \Bbb Q(x_i) $ is cyclic with degree $2^i$ Well.. at ...
7
votes
1answer
2k views

Existence of irreducible polynomials over finite field

Let $F$ be a finite field. How do we prove that for each $n \in \mathbb{N}$ there is an irreducible polynomial of degree $n$? One can assume that $F = \mathbb{F}_{p^m}$ where $p$ is prime. If $n \ge ...
7
votes
2answers
330 views

Is a completion of an algebraically closed field with respect to a norm also algebraically closed?

Assume we have an algebraically closed field $F$ with a norm (where $F$ is considered as a vector space over itself), so that $F$ is not complete as a normed space. Let $\overline F$ be its completion ...
7
votes
2answers
163 views

If $E/F$ is algebraic and every $f\in F[X]$ has a root in $E$, why is $E$ algebraically closed?

Suppose $E/F$ is an algebraic extension, where every polynomial over $F$ has a root in $E$. It's not clear to me why $E$ is actually algebraically closed. I attempted the following, but I don't think ...
7
votes
3answers
172 views

Every element is radical in a field extension.

Let $L/K$ be an algebraic field extension. Suppose for each $x\in L$, there exists an integer $n>0$ such that $x^n\in K$, where $n$ may depend on $x$. If the characteristic of $K$ is zero, does it ...
7
votes
2answers
174 views

Is it true that every element of $\mathbb{F}_p$ has an $n$-th root in $\mathbb{F}_{p^n}$?

It is not hard to prove that every element of $\mathbb{F}_p$ has a square root in $\mathbb{F}_{p^2}$: take any $a \in \mathbb{F}_p$ and consider the polynomial $f = X^2 - a$. If $f$ has a root in ...
7
votes
2answers
130 views

Showing $\mathbb{Q}(2^{1/3}+2^{1/5})=\mathbb{Q}(2^{1/3},2^{1/5})$

In order to prove $[\mathbb{Q}(2^{1/3}+2^{1/5}):\mathbb{Q}]=15$, I want to show $\mathbb{Q}(2^{1/3}+2^{1/5})=\mathbb{Q}(2^{1/3},2^{1/5})$. Any suggestions?
7
votes
1answer
301 views

Generating Elements of Galois Group

I am trying to prove the following: Let $K=\mathbb{Q}(\sqrt{p_1},\ldots,\sqrt{p_n})$. Show that $K/\mathbb{Q}$ is Galois with Galois group $(\mathbb{Z}/2\mathbb{Z})^n$. I have attached my proof ...
7
votes
1answer
477 views

An exercise on cyclic extensions of Hungerford's book, Algebra.

I'm trying to show the following exercise of the book, it is in the section of cyclic extensions and says the following: Let $\overline{\mathbb Q}$ be a fixed algebraic closure of $\mathbb Q$, let ...
7
votes
1answer
226 views

How to show that $(\mathbb{Q}(\sqrt{2}))^{\times} \cong (\mathbb{Q}(\sqrt{3}))^{\times}$

How to show that $$(\mathbb{Q}(\sqrt{2}))^{\times} \cong (\mathbb{Q}(\sqrt{3}))^{\times}$$ where $(\mathbb{Q}(\sqrt{2}))^{\times}$ is multiplicative group of $\mathbb{Q}(\sqrt{2})$. Mapping ...
7
votes
2answers
266 views

Splitting field of $x^{13}+1$ over $\mathbb{Q}$

I want to know how to calculate the degree of the field extension $[K:Q]$ where $K$ is the splitting field of $x^{13}+1$ over $\mathbb{Q}$. I'm new to this area and this is not really covered in my ...
7
votes
1answer
250 views

Algebraic Extensions and Separability

I have been wrestling with the following problem for the past few hours, but I have made no progress whatsoever, namely: Let $K/k$ be an algebraic extension with characteristic $p>0$ and let ...
7
votes
1answer
412 views

Can a field be isomorphic to its subfield?

Let $K$ be a field and $K(X)$ be the field of its rational functions. Now let $\phi \in K(X)$ be a rational function such that $K(\phi) \neq K(X)$. Now, since $\phi$ is transcendental over $K$, ...
7
votes
2answers
209 views

Application of the Artin-Schreier Theorem

This is exercise $6.29$ out of Lang's book: Let $K$ be a cyclic extension of a field $F$, with Galois group $G$ generated by $\sigma$. Assume that the characteristic is $p$, and that ...
7
votes
1answer
127 views

Lower Bound on Number of Wildly Ramified Extensions of $\mathbb{Q}_p$

Suppose $p$ is prime and $p$ divides $e$ divides $n$. Typically up to isomorphism there are a lot of wildly ramified extensions of $\mathbb{Q}_p$ which have degree $n$ and ramification index $e$. ...
7
votes
2answers
332 views

Compositum of fields with trivial intersection

Let $E/F$ be a finite extension. Let $L,K$ be two intermediate fields with $L\cap K = F$, and also $$[L : F] [K:F] = [E:F].$$ Must it hold that the compositum $LK$ equals $E$? If we assume that $E/F$ ...
7
votes
1answer
129 views

“Real part” of a number field

Let ${\mathbb K} \subseteq {\mathbb C}$ be a finite extension of $\mathbb Q$, and let $n=[{\mathbb K} : {\mathbb Q}]$. Let $X_{\mathbb K}$ denote the set of all “components” (i.e., real and imaginary ...
7
votes
1answer
155 views

Exercise on finite intermediate extensions

Let $E/K$ be a field extension, and let $L_1$ and $L_2$ be intermediate fields of finite degree over $K$. Prove that $[L_1L_2:K] = [L_1 : K][L_2 : K]$ implies $L_1\cap L_2 = K$. My thinking ...
7
votes
1answer
564 views

$K/E$, $E/F$ are separable field extensions $\implies$ $K/F $ is separable

I wish to prove that if $K/E$, $E/F$ are algebraic separable field extensions then $K/F$ is separable. I tried taking $a\in K$ and said that if $a\in E$ it is clear, otherwise I looked at the minimal ...
7
votes
2answers
646 views

How to find irreducible polynomials over $\mathbb{Q}(i)$ with prescribed Galois group?

Here is my recent homework question: For each of the following five fields $F$ and five groups $G$, find an irreducible polynomial in $F[x]$ whose Galois group is isomorphic to $G$. If no example ...
7
votes
1answer
199 views

Calculating Separable Closures

In my study of fields, the notion of the separability of an algebraic field extension is one of the more slippery concepts I have encountered thusfar. What is particularly vexing to me is the notion ...
7
votes
1answer
91 views

Counting diagonalizable matrices in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$

How many diagonalizable matrices are there in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$ ? Where $p$ is a prime number. Attempt : By definition a matrix is called diagonalizable if there exists an ...
7
votes
1answer
128 views

Multivariate coprime polynomials in field extensions

Suppose $f$ and $g$ are polynomials in $n$ variables, $n\ge 2$, over a field $E$. Suppose further that $f$ and $g$ are relatively prime over $E$. If $F$ is a field extension of $E$, are $f$ and $g$ ...
7
votes
1answer
113 views

Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.

Do you know how to compute a uniformizer of $\mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})$? Where $\zeta_{p^n}$ is a primitive $p^n$-th root of 1 and $p$ is an odd prime.
7
votes
1answer
125 views

Should this be viewed as a serious issue with the meadow-theoretic approach?

Meadow theory (see here) allows us to apply the results and concepts of universal algebra to the study of fields. Obviously, this is very, very nice. However, I have the following issue with the ...
7
votes
2answers
388 views

How does $\cos(2\pi/257)$ look like in real radicals?

We know $\cos(2\pi/p)$ for p a Fermat prime can be expressed in real radicals. The case $p=17$ is a root of an 8th deg eqn, but can be also given as a sequence of nested radicals, $$\begin{aligned} ...
7
votes
1answer
230 views

Show the two fields are not isomorphic

Let $p,q,r$ be prime integers with $q\neq r$. Let $\sqrt[p]q$ denote any root of $x^p-q$ and $\sqrt[p]r$ denote any root of $x^p-r$. Please show that $\mathbb{Q}(\sqrt[p]q)\neq\mathbb{Q}(\sqrt[p]r)$. ...
7
votes
1answer
241 views

Two questions on Nagata's counterexample to the Hilbert's fourteenth problem.

Let $\{a_{ij}\}$ for $i=1,2,3$, and $j=1,...,16$ be algebraically independent elements over some prime field. Let $k$ be a field containing all $a_{ij}$. Then consider $k^{16}$ as $k$-vector space and ...
7
votes
1answer
185 views

the number field $\mathbb{Q}(\cos \frac \pi n)$

Let $x$ be $\cos \displaystyle \frac \pi n$ for some natural number $n$. Then is it true that $\mathbb{Q}(x^2+x)=\mathbb{Q}(x)$?
7
votes
1answer
270 views

Extension of Homomorphisms (Lang, Atiyah and McDonald)

Let $A$ be a subring of a field $K$, and suppose that $A$ is a local ring with maximal ideal $\mathfrak{m}$. Let $x \in K, \, x \neq 0$. Let $\phi: A \rightarrow L$ be a homomorphism of $A$ into the ...
7
votes
1answer
333 views

Problem 18.7 in I. Martin Isaacs' Algebra

I am trying to solve the following problem in I. Martin Isaacs' Algebra: A graduate course, p.290: Let $f(X),g(X) \in F[X]$ and suppose $E \supseteq F$ is the splitting field both for $f(X)$ and ...
7
votes
1answer
168 views

If $K=K^2$ then every automorphism of $\mbox{Aut}_K V$, where $\dim V< \infty$, is the square of some endomorphism.

I have to show the following: Let $K$ be a field such that $\mbox{char } K \neq 2$ and each element of $K$ is a square (i.e. $K^2=K$) and let $V$ be a finite-dimensional vector spaces over $K$. ...
7
votes
0answers
84 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
7
votes
1answer
602 views

Patrick Morandi- Field and Galois Theory- Section 4- Exercise 11

Let $K$ be the rational function field $k (x)$ over a perfect field $k$ of characteristic $p > 0$. Let $F = k(u)$ for some $u$ in $K$, and write $u = \frac {f(x)}{g(x)}$ with $f$ and $g$ ...
6
votes
4answers
700 views

Does there exist a field which has infinitely many subfields?

Does there exist a field which has infinitely many subfields? Does there exist an enormous supply of such fields? I don't know how to begin.
6
votes
3answers
474 views

Is $\mathbb{Q}[\sqrt2]$ = $\mathbb{Q}[\sqrt2+1]$?

Is $\mathbb{Q}[\sqrt2]$ = $\mathbb{Q}[\sqrt2+1]$? I think so because $$\mathbb{Q}[\sqrt{2}+1] = \{\sum_{i=0}^{n}c_i(\sqrt{2}+1)^i\mid n\in\mathbb{N}, c_i\in\mathbb{Q}\}$$ $$= ...