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)

3
votes
1answer
88 views

Why is $[\mathbb{Q}(e^{2\pi i/p}):\mathbb{Q}] = p-1 $?

I know that $e^{2\pi i/p}$ is a root of $x^{p}-1$ and we can write: $ x^{p}-1=\left(x-1\right)\left(\sum_{i=0}^{p-1}x^{i}\right) $ So $e^{2\pi i/p}$ is a root of $\sum_{i=0}^{p-1}x^{i}$, which means ...
1
vote
1answer
219 views

A question about the degree of an element over a field extension.

Say $K$ is a field extension of field $F$. If element $b$ is algebraic with degree $n$ over $F$, we know that $[F(b):F]=n$. Why is it that $[K(b):K]\leq n$?
1
vote
2answers
52 views

A doubt regarding nature of $F(a)$, where $F(a)$ is the intersection of all fields containing field $F$ and element $a$.

Proof in Herstein: Let us consider elements like $f_{0}+f_{1}a+f_{2}a^{2}+\dots f_{s}a^{s}$. Here $f_{0},f_{1}\dots f_{s}\in F$. Now consider the quotient field $U$ generated by elements like ...
6
votes
4answers
154 views

Strong characterization of $\mathbb C$ with respect to $\mathbb R$

$\mathbb R^2$ is not a field, but $2$-tuple arithmetic rules like $(a,b)(c,d)=(ac-bd,ad+bc)$ coupled with $\mathbb R^2$ make it a field, but are there other rules than $(a,b)(c,d)=(ac-db,ad+bc)$ ...
0
votes
0answers
61 views

A question about field extensions

I read in Herstein that a field extension $K$ of field $F$ is a vector space over $F$, and its degree over $F$ is the number of base elements. Let $K$ (field extension of $F$) be a vector space over ...
2
votes
0answers
67 views

Field extension

I need to prove that if $F$ is a field and $u=\frac{f(t)}{g(t)} \in F(t)$ (where $f,g$ are coprime in $F[t]$) then $[F(t):F(u)]=\max(\deg f,\deg g)$. I know I have to prove that $ug(x)-f(x)$ is ...
0
votes
3answers
56 views

automorphisms and field extension $E$ of $\mathbb{Q}$.

I want a hint. That is all I ask for. The question I am asked to prove is as follows: Let $E$ be an extension field of $\mathbb{Q}$. Show that any automorphisn of $E$ acts as the identity on ...
7
votes
1answer
91 views

Given $G$, when can we find a division ring $R$ with $R^*=G$?

This is motivated by a characterization of finite cyclic groups, in which one proves Let $G$ be a finite group. If $\#\{g\in G\colon g^d=e\}$ is at most $d$, then $G$ is cyclic. The proof is ...
3
votes
1answer
266 views

Finite Extensions and Roots of Unity

Two questions; the hint I've been provided is that they are, in fact, related. Prove that a finite extension of $\mathbb{Q}$ contains finitely many roots of unity. What is the largest (finite) ...
2
votes
2answers
29 views

Relationships of Eigenvalues in Algebraic Closure

Suppose that $k$ is a field, and $A \in M_n(k)$ is a matrix that becomes diagonalizable over $\overline{k}$, the algebraic closure of $k$. Let $\lambda_1, \ldots, \lambda_n$ denote the (not ...
6
votes
0answers
249 views

Is there a general algorithm to find a primitive element of a given finite extension (with a finite number of intermediate fields)?)

I just want to know if there is an algorithm to find the primitive element of a given finite extension $F/k$ if the intermediate fields are given. I know how to approach it in particular examples ...
2
votes
0answers
174 views

Non-isomorphic simple extensions of the same degree of a field of positive characteristic

Let $K$ be a field of positive characterstic. I need to show that there exists $a$ and $b$ of the same degree over $K$ but $K(a)$ and $K(b)$ are not isomorphic. I thought of an example where they are ...
11
votes
0answers
248 views

Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a ...
1
vote
0answers
140 views

$X^n-a \in k[X]$ ,char(k)|n. Multiplicity of roots of irreducible polynomial which devides $X^n-a$.

Let $k$ be a field of $char(k)=p>0$, $f(X)=X^n-a \in k[X](a \neq 0)$, $p|n$. If $g(X)$ is an irreducible polynomial in $k[X]$ and $g(X)|f(X)$, do all the roots of $g(X)$ have the same ...
3
votes
0answers
104 views

(Revised) Does p(a)=p(b) => a=b?

This is a revised version of problem posted here named :Does $p(a) = p(b) \Rightarrow a=b \ $? Let $S=(P_1,P_2,…,P_n)$ be a set of polynomials with complex coefficients. I call S critical if it ...
0
votes
1answer
382 views

Proving an Integral domain is a field. [duplicate]

Let $R$ be an integral domain containing a field $F$ as a subring. Show that if $R$ is a finite dimensional vector space over $F$, then $R$ is a field. This is a Ph.D. entrance question, I recently ...
5
votes
2answers
110 views

Galois Extensions and $n^{\text{th}}$ Roots

I've been studying for my prelims lately, and this problem has me stuck: (a) Let $K$ be a field with no abelian Galois extensions. Suppose that $n$ is a positive integer and either $char(K)=0$ or ...
5
votes
1answer
120 views

finding fixed field of automorphism

Let $F$ be a field and let $g:F(x) \to F(x)$ be the automorphism which maps $x$ to $x+1$. I need to find the fixed field of this automorphism. So far I know $g$ fixes $F$. I want to use Galois ...
2
votes
1answer
475 views

Galois group of $x^8+2$ over $\Bbb{Q}$

This is what I did to find the Galois group for $x^8+2$: Splitting field: $$K = \Bbb{Q}(\zeta_8, \zeta_{16}2^{1/8})$$ Since $Gal(\Bbb{Q}(\zeta_8)/\Bbb{Q}) \cong Aut( \langle\zeta_8\rangle) \cong ...
2
votes
1answer
273 views

Calculating The Galois Group of the Splitting Field of $f=x^3-3$

If we let $f=x^3-3$ the let L be the splitting field for this polynomial I am trying to find $\Gamma(L:\mathbb{Q})$ and all intermediate field extensions. Now as this is a splitting field and finite ...
2
votes
1answer
200 views

Characteristic of commutative semisimple rings?

In one of my questions (Structure of the group ring of a direct product?), a statement is made for a commutative semisimple ring of characteristic $p^t, t\geq1$. Now I don't understand why there ...
1
vote
1answer
56 views

minimal polynomial of an element

I we have that $\alpha$ is the positive real root of $f=x^4-3$ then the splitting field of $f$ over $\mathbb{Q}$ is $\mathbb{Q}(\alpha,i)$ if I then want to find the degree of the extension of this ...
3
votes
2answers
116 views

Show $\zeta_p \notin \mathbb{Q}(\zeta_p + \zeta_p^{-1})$

I asked a question here: [Writing a fixed field as a simple extension of $\mathbb{Q}$ ], but realised I couldn't justify why the given quadratic was irreducible. Thus: Is there a way of showing ...
2
votes
2answers
267 views

characteristic vs minimal polynomial

Let $L$ be a finite field extension of $K$. For every element $\theta$ in $L$ define the characteristic polynomial of $\theta$ as follows $$\operatorname{char}_{\theta}(X):=\det(X\cdot ...
19
votes
0answers
450 views

When are nonintersecting finite degree field extensions linearly disjoint?

Let $F$ be a field, and let $K,L$ be finite degree field extensions of $F$ inside a common algebraic closure. Consider the following two properties: (i) $K$ and $L$ are linearly disjoint over $F$: ...
2
votes
1answer
134 views

A basic question on factorization

Is the following true? If not, can anyone add some reasonable assumptions to make it true? Statement. Let $E/F$ be a field extension and let $\alpha \in E \setminus F$ be algebraic over $F$. If $f(x) ...
1
vote
1answer
47 views

Field, Euclidean division question.

Let $K$ be a field and $f \in K[x]$. Show that if there is some $a \in K$ such that $f(a)=0$, then $x-a$ divides $f$. My friend told me to use Euclidean division by $x-a$. Also show that a ...
1
vote
0answers
46 views

A question regarding linear disjiontness and the degree of a field extension

Let $K / k$ and $L / k$ be field extensions with $K$ and $L$ linearly disjoint over $k$. Suppose that $K' / K$ is an extension of finite degree, and that all fields are characteristic zero. Then is ...
1
vote
2answers
290 views

Help with proof concerning ordered fields and smallest elements

I have been trying to prove that every ordered field has no smallest positive element for a while now, and I think I have it worked out. However, I feel like something is missing in my proof. Here is ...
5
votes
4answers
325 views

Let $K$ be a subfield of $\mathbb{C}$ not contained in $\mathbb{R}$. Show that $K$ is dense in $\mathbb{C}$.

Let $K$ be a subfield of $\mathbb{C}$ not contained in $\mathbb{R}$. Show that $K$ is dense in $\mathbb{C}$. completely stuck on it. can I get some help please.
1
vote
0answers
56 views

Transitive action on the set of algebra homomorphisms.

Let $k$ be a field, and $K/k$ be a Galois extension. Suppose $K'/k$ be an extension with $K'$ is a finitely generated $k$-algebra. Then the Galois group $\textrm{Gal}(K/k)$ acts canonically on the set ...
1
vote
1answer
79 views

Solve equations in a field with characteristic p.

Let $p$ be a prime and $K$ a field with characteristic $p$. How to solve the equation $x^2+2=0$ in the field $K$? Here $x, 2, 0 \in K$. Is it equivalent to solve the equation $x^2+2 = 0 \pmod p$? What ...
0
votes
1answer
165 views

A question regarding the finiteness of the degree of a field extension

Let $x$ and $y$ be transcendental and $y$ be algebraic over $\mathbb{Q}(x)$. Let $F$ be an algebraic Galois extension of $\mathbb{Q}(x)$ of infinite degree and let $\mathbb{Q}^{al}$ be the algebraic ...
1
vote
2answers
105 views

Question in Hungerford regarding field extensions

In Algebra by Hungerford, page 237 the sketch of proof for Theorem 1.10: Theorem 1.10: If $K$ is a field and $f\in K[x]$ polynomial of degree $n$, then there exists a simple extension field $F = ...
-1
votes
4answers
114 views

Allowing the zero element in a field to have an inverse

In the definition of a field one of the required properties is that every element other than zero has a multiplicative inverse. It's vague whether the zero is forced not to have an inverse or not, ...
1
vote
2answers
83 views

Nonreal units in totally imaginary number fields

Suppose we are given a totally imaginary number field $L$ of degree greater $2$, is it possible that all units of $L$ lie in $\mathbb R$? This is true for almost all quadratic imaginary number fields, ...
4
votes
2answers
163 views

3 questions on field extensions

I am trying to figure out some things regarding field extensions and some questions have arisen on the way. Let $a$ be a positive integer which doesn't have a rational $nth$ root: Is the splitting ...
5
votes
3answers
1k views

Quadratic subfield of cyclotomic field

Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be ...
3
votes
1answer
105 views

Why don't I end up with the same splitting field?

I've understood that the splitting field of $x^4+2$ and the splitting field of $x^4-2$ over $\mathbb{Q}$ are both the field $\mathbb{Q}(\sqrt[4]{2} , i)$. With degree $8$ over $\mathbb{Q}$. This ...
6
votes
3answers
224 views

Is there a pattern of the factorization of a polynomial modulo $p$ as $p$ varies

Take $P\in\mathbb{Z}[X]$ and factorize it modulo $p$, where $p$ is a prime. Modulo different $p$'s the factorization varies. Is there a pattern in this variation? I mean, for example, if $P$ is ...
2
votes
1answer
154 views

Splitting field for $x^n+a$

What is a splitting field $E$ for $f(x)=x^n+a$ over the field $K$ of characteristic zero? If I put $g(x)=x^{2n}-a^2=(x^n-a)(x^n+a)$. The splitting field $F$ of $g(x)$ is $K(\sqrt[n]{a},\alpha )$ ...
3
votes
1answer
190 views

Galois group of irreducible quartic with real coefficients

Let $K$ be a subfield of the real numbers and $f\in K[x]$ be an irreducible quartic. If $f$ has exactly two real roots, show that the Galois group of $f$ is $S_{4}$ or $D_{4}$ (I'm using the ...
3
votes
1answer
143 views

Irreducible polynomial over $\mathbb Z/7\mathbb Z$

Is the polynomial $t^6-3$ irreducible over $\mathbb{Z}/7\mathbb{Z}$? How would you prove that without doing all the computations? Is there a general method for this? Thank you.
1
vote
2answers
91 views

Example where $[E:\mathbb{Q}]<|\mathrm{Aut}_{\mathbb{Q}}E|$

Let $F$ be an algebraic closure of $\mathbb{Q}$ and let $E\subset F$ be a splitting field over $\mathbb{Q}$ of the set $\{x^{2}+a|a\in\mathbb{Q}\}$ so that $E$ is algebraic and Galois over ...
0
votes
1answer
69 views

Smallest Galois extension

Let $F$ be a finite dimensional Galois extension of $K$ and let $E$ be an intermediate field. Show that there is a unique smallest field $L$ such that $E\subset L\subset F$ and $L$ is Galois over $K$. ...
2
votes
1answer
158 views

Showing that a field extension is Galois

Let $F$ be an extension field of a field $K$. Let $E$ be an intermediate field such that $E$ is Galois over $K$, $F$ is Galois over $E$, and every $\sigma\in\mathrm{Aut}_{K}E$ is extendible to $F$. ...
8
votes
1answer
122 views

Algebraic closure of $(\mathbb{Z}/p\mathbb{Z})(T)$

Is there a concrete description of the algebraic closure of $(\mathbb{Z}/p\mathbb{Z})(T)$?
0
votes
1answer
141 views

Sum of the Hyperharmonic\Over-harmonic Series under $\mathbb{Z}_n$ for $p=2$

For $n \geq 5$ prime number, calculate the sum of: $$1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{(n-1)^2}$$ under $\mathbb{Z}_n$. I figured it's the hyperharmonic\over-harmonic series, $$ ...
3
votes
5answers
81 views

On any finite field, adding the identity element a finite amount of times will result to the neutral element

Show that if you add the identity element ($1$) a finite amount of times will result to the neutral element ($0$). I started with saying that in every field there's an element $a \in F$ so that $a + ...
0
votes
2answers
294 views

Addition table for a 4 elements field

Why is this addition table good, \begin{matrix} \boldsymbol{\textbf{}+} & \mathbf{0} & \boldsymbol{\textbf{}1} & \textbf{a} &\textbf{ b}\\ \boldsymbol{\textbf{}0} & 0 & 1 ...