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 topic....

learn more… | top users | synonyms (1)

0
votes
1answer
30 views

$\mathbb R[x]/\langle x^2+1\rangle\simeq\mathbb C$ and $\mathbb R[x]/\langle x^2+1\rangle$ contains a zero of $x^2+1$

How to show that $\mathbb R[x]/\langle x^2+1\rangle\simeq\mathbb C$ and $\mathbb R[x]/\langle x^2+1\rangle$ contains a zero of $x^2+1$ Due to the division algorithm, $\mathbb R[x]/\langle x^2+1\...
2
votes
2answers
317 views

Show that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}$ is not Galois; prove that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}(2^{1/2})$ is Galois

I would like to show that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}$ is not Galois. Can I just say that it is not separable because $2^{1/4} \in \mathbb{Q}(2^{1/4})$ but its minimal polynomial in $\mathbb{Q}$ ...
-1
votes
1answer
41 views

Permutation of a fixed field is an intermediate field corresponding with the conjugate of the group corresponding to the fixed field

The following is my question: Let $K/F$ be a Galois extension with Galois group $G = Gal(K/F)$, with intermediate field $L: F \subseteq L \subseteq K$ which corresponds to subgroup $H \leq G$ by the ...
2
votes
1answer
380 views

Every Intermediate Field of Abelian Galois Field Extension is Splitting Field of a Separable Polynomial

This is my question: Suppose the $K/F$ is a Galois extension with an abelian Galois group $G$. Prove that every intermediate field $L: F \subseteq L \subseteq K$ is the splitting field (over $F$) of ...
1
vote
0answers
132 views

Union of field extensions over Q

I am asked to prove that $L=\bigcup_{n=1}^\infty\mathbb{Q}(\sqrt[n]2)$ is an algebraic field extension over $\mathbb{Q}$. So far I have: Let $\beta\in L$, then by definition of union there exists a $...
2
votes
2answers
124 views

Prove the ring $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ has inverse and is a field

How can I prove that $\frac{1}{a+b\sqrt[3]{2}+c\sqrt[3]{4}}$ is of the form $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ (i.e. that $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ is a field) for all rational $a,b,c$ and $a+b\sqrt[3]...
3
votes
1answer
80 views

Algorithm to find representation of an element of field extension $\mathbb{Q}(q)$ in the form $\sum a_i q^i$

Let $\mathbb{Q}(q)$ be a field extension of $\mathbb{Q}$, where $q$ is a real root of some monic irreducible polynomial $p(x) \in \mathbb{Z}[x]$ of degree $d=3$. Given $x \in \mathbb{R}$, (or $\...
2
votes
1answer
161 views

Why characters are continuous

According to Wikipedia: ''Every character is automatically continuous from $A$  to $\mathbb C$, since the kernel of a character is a maximal ideal, which is closed. '' where $A$ is a Banach algebra. ...
0
votes
0answers
56 views

Need some help finishing this proof about characters in Banach algebras

I tried to prove: Let $A$ be a commutative unital complex Banach algebra. Then there is a bijection between the maximal ideals in $A$ and the set of non-zero homomorphisms $A \to \mathbb C$. But I ...
3
votes
1answer
54 views

Number of isomorphisms between two fields

Let $F,F'$ be two fields. Is there anything that can be said about the number of isomorphisms that can exist? In particular can there be more than one? What if $F$ is the complex numbers $\mathbb C$? ...
1
vote
2answers
89 views

When does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$?

Let $p$ be a prime integer, and let $q=p^r$ and $q'=p^k$. For which values of $r$ and $k$ does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$? From Artin's Algebra, Chapter 15, problem 7.12 from the ...
2
votes
1answer
84 views

Why $K(u)$ is a field?

Let $F$ be an extension field of $K$ and $u\in F$. How do we know that adjoining an element of F to K, makes $K(u)$ a field? I know that $Q(\sqrt2)=\{a+b\sqrt2|a,b\in Q\}$ is a field, but in the ...
0
votes
3answers
263 views

Proof about finite dimensional vector spaces over fields

Prove that every finite dimensional vector space $V $of dimension $n$ over a field $F$ is isomorphic to the vector space $F^n$. Okay, lot's of stuff here. I think most of the reason I can not do this ...
2
votes
2answers
125 views

Ring homomorphism with field as image, is the pre-image also a field?

Let $f:R\rightarrow S$ be a surjective ring homomorphism. $R,S$ are both integral domains. Suppose $S$ is a field, then is $R$ also a field? A possible useful fact: A finite integral domain is a ...
1
vote
2answers
71 views

Analogy between the quaternion ring and extensions of the rationals

I've started studing fields and their extensions. As an exercise I proved that $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$ by showing that $B=(1,\sqrt2,\sqrt3,\sqrt6)$ is a base for the extension field ...
3
votes
1answer
125 views

Find primitive element of splitting field of $1 + x + x^2 - x^5$

As the title says, I need to find the primitive element of the splitting field of $1 + x + x^2 - x^5$ over $\mathbb{Q}$. Firstly, I would proceed by finding the roots as the splitting field has to ...
0
votes
3answers
32 views

Ring theory question.

Why is a field with 27 elements has characteristic 3? I was solving a question and I came to know this fact which I didn't know before. Is there anyone who can explain this to me? Thanks in advance.
2
votes
2answers
58 views

Constructing expressions

Suppose you are given all the elementary symmetric functions of $n$ variables $x_1,x_2,...,x_n$ and two rational functions $A(x_1,x_2,...,x_n)$ and $B(x_1,x_2,...,x_n)$ in the same $n$ variables that ...
2
votes
2answers
265 views

Field extension of a finite field

Let $E$ be an extension field of a finite field $F$ , where $F$ has $q$ elements. Let $a \in E$ be algebraic over $F$ of degree $n$. Prove that $F(a)$ has $q^n$ elements. I am not sure how to do this ...
3
votes
2answers
96 views

Group-Isomorphism problem

I want to find an group-isomorphism $$ \psi : (\mathbb{Z}/8\mathbb{Z},+) \longrightarrow \mathbb{F}_9^\times $$ which should be used to multiply elements in $\mathbb{F}_9$ or to find the inverse ...
0
votes
1answer
115 views

Extension Fields and Quotients

In the Dummit and Foote 3ed chapter on field extensions (ch. 13), it is stated as a theorem (6) that $ F(\alpha) \cong F[x]/(p(x))$ where $\alpha$ is a root of $p(x)$ and goes on to state that any ...
4
votes
2answers
684 views

Field extensions and algebraic/transcendental elements

Let $E$ be an extension of field $F$, and let $\alpha, \beta \in E$. Suppose $\alpha$ is transcendental over $F$ but algebraic over $F(\beta)$. Show that $\beta$ is algebraic over $F(\alpha)$. Okay,...
1
vote
1answer
56 views

Splitting field in $\mathbb{C}$ over $\mathbb{Q}$

I want to find the splitting field in$\mathbb{C}$ of $x^4-4$ over $\mathbb{Q}$. I solved for the zeros, which is $i\sqrt2, -i\sqrt2, \sqrt2, -\sqrt2$, so the splitting field, say $E$, is just $\...
9
votes
2answers
584 views

$\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})=\mathbb{Q}$?

Is there an easy way to see that $$\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})=\mathbb{Q}?$$ I know that $\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})$ is a subfield of $\mathbb{Q}(\sqrt[...
1
vote
1answer
59 views

Computing a field extension by hand

Let $k$ be a field, and $K=k(t)$ be the field of rational functions (where $t$ is indeterminate). Let $F=k(t^2)$. A typical element of $F$ will look like: $$ \frac{\displaystyle \sum_{i=0}^{n} a_{2i} ...
5
votes
0answers
73 views

The reducibility of polynomial $x^{2^n}+x+1$ over $Z_2$ [duplicate]

I meet the problem in Hungerford's Algebra, page 282 as follows: If $n>2$, then the polynomial $x^{2^n}+x+1$ is reducible over $Z_2$ . I have no any idea on this. But I really want to know how to ...
1
vote
1answer
51 views

Splitting field in multiple field extension

Let $F\subset K\subset E$ be fields, $f(x)\in F[x]$. E is the splitting field of $f(x)$ over $F$. Prove $E$ is the splitting field of $f(x)$ over $K$. My problem with this proof is that I do not know ...
5
votes
0answers
162 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like "intuitively..."...
1
vote
1answer
107 views

Splitting field of a set of separable polynomials implies separability of extension.

Let $F$ be a splitting field of $S\subset K[x]$ over $K$, where $S$ is a set of separable polynomials. I want to show that $F$ is separable over $K$, meaning for all $u\in F-K$, the irreducible ...
0
votes
2answers
52 views

Prove: 2^(1/3) cannot be written in terms of any given root of unity

How does one prove that there exists no natural number $n$ such that $\sqrt[3]{2} $ belongs to the field extension of the rationals by the $n$th root of unity?
0
votes
1answer
83 views

Operator generating subuniverse generated by X is algebraic closure operator

This is taken from Universal Algebra Text book by Stan Burris. I have a question regarding the last conclusion as to how does the author conclude that Sg is an algebraic closure operator. How do we ...
2
votes
2answers
75 views

Infinite field extension but algebraic subset

Is there a subset $ S \subseteq \mathbb{C}$ so that all $s \in S$ are algebraic over $\mathbb{Q}$ and $\left[\mathbb{Q}[S] \colon \mathbb{Q} \right]=\infty$? $\mathbb{Q}[S]$ is defined as $\mathbb{Q}[...
1
vote
1answer
80 views

How do we explain the existence of complex conjugation?

We can define the complex numbers by writing $\mathbb{C} = \mathbb{R}[i]/(i^2+1),$ where $\mathbb{R}$ is to be regarded as a commutative ring. Furthermore, since $\mathbb{R}$ happens to be a field, ...
0
votes
0answers
40 views

For which $d\in\mathbb{Z}$, $\mathbb{Q}(\sqrt{d})$ primitive root of unity of order $p>2$ prime

If $p>2$ is a prime number, then I have to find $d\in\mathbb{Z}$ such that we have a primitive root of unity of order $p$. I know that $d<0$ because otherwise, you can never have a root of unity....
1
vote
1answer
36 views

Why is every monomorphism of E into $\overline{\mathbb{Q}}$ a $\mathbb Q$-monomorphism of E into $\overline{\mathbb{Q}}$?

I've been trying to solve the problem below, but I'm not even sure how to get started. Any help would be greatly appreciated. I feel like there is a key insight that will solve the problem, but I'm ...
7
votes
1answer
185 views

Show from the axioms: Addition in a quasifield is abelian

According to wikipedia a quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is a group. (As usual, we denote its identity element by $0$.) $(Q\setminus\{0\},\cdot)$ is a loop. (Its ...
3
votes
1answer
137 views

Can't find intermediate subfield of these cyclotomic extensions!

I'm to look at $\mathbb{Q}(\zeta_7)$ and $\mathbb{Q}(\zeta_{10})$, and they both have a common thing I can't see how to work around. For $\mathbb{Q}(\zeta_7)$, skipping some details I doubt matter ...
2
votes
1answer
89 views

Galois group of intermediate fields

Find all subfields $M \subset \mathbb{Q}(\zeta_{7})$ where $\zeta_7=e^{2\pi i/7}$ and determine $Gal(\mathbb{Q}/M)$ and $Gal(M/\mathbb{Q})$. I've found that there are 2 intermediate fields $L=\mathbb{...
4
votes
0answers
825 views

Proving that the algebraic reals form an infinite field extension over $\mathbb Q$

I believe I have proved the following, from the free textbook here, page 356. I have 2 different proofs and would greatly appreciate thoughts on them. Show that the set of all elements in $\mathbb{...
1
vote
0answers
14 views

Why can an ideal generated by a subset be written in this form?

I have a subset $F \subset R$ that generates an ideal $(F)$. Apparently this can be written in the form $$(F)=\{a_1f_1b_1+...+a_kf_kb_k|k \ge 0, f_i \in F, a_i,b_i\in R\}$$ or if $R$ is commutative ...
0
votes
1answer
21 views

Tower rule for transcedence degree

$\newcommand{\tr}{\operatorname{tr}}$ Let $k \subset E \subset K$ be extension fields. Show that $$\tr\ \deg (K/k)= \tr\ \deg (K/E) + \tr\ \deg (E/k).$$ Here if we consider $S_1$ and $S_2$ be two ...
1
vote
2answers
125 views

Does $x\cdot 0 = 0$ follow from the field axioms alone?

From the field axioms alone, does it follow that $x \cdot 0 = 0$ for all $x$? All I would like is a statement that it can or cannot be done (hints not necessary). I would like to do it myself; I ...
0
votes
1answer
139 views

Is the residue field of an algebraically closed field with respect to a non-trivial valuation infinite?

Let $K$ be an algebraically closed field with a non-trivial valuation. Let $\Bbb k$ be the residue field, i.e. $\Bbb k = R/\mathfrak m$ where $R$ is the valuation ring $R:=\{a\in K\mid \text{val} (a)\...
8
votes
2answers
341 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 ...
2
votes
2answers
107 views

Reducibility of a Cyclotomic Polynomial under the ring homomorphism $\mathbb{Z} \rightarrow \mathbb{F}_p$

I'm working through the following question: Question Reference: Oxford Part I Paper B2 2003 Find the monic polynomial $f \in \mathbb{Z}[X]$ whose roots are the complex primitive $12^{\...
2
votes
2answers
220 views

Question About Notation In Field Theory.

I have a question about notation specifically square brackets $[$ and round brackets $($. My textbook doesn't explain any of this and I cannot find a reliable source online to confirm the difference. ...
2
votes
1answer
98 views

Constructing splitting fields

Construct the splitting field for the polynomial $(x^4 - x^2 - 2)$ over $\mathbb Q$ (rationals) . What is the degree of the extension? Why? How would one go about tackling this question? I'm a bit ...
4
votes
1answer
162 views

finite field extension problem

Maybe somebody knows how to proove the following algebraic theorem: $C \subset U$ is a field extension and $N \subset U$ so, that all $x \in N$ are algebraic over $C$ and $C[N]=\left\lbrace f(x_1,\...
2
votes
1answer
84 views

Separable field extensions

Let $k$ be a field and $k(x_1,x_2,...,x_n)= k(x)$ a finite separable extension. Let $u_1,u_2,..., u_n$ be algebraically independent over $k$. Let $w= u_1x_1 + u_2 x_2 +\cdots +u_n x_n .$ Let $k_u = k(...
1
vote
1answer
60 views

The fixed field of a galois group if the characteristic is $p$.

Consider the following proposition with its relative proof: Let $k$ be an algebraically closed field of characteristic $0$. a) If $L$ is a subfield of $k$, then every elements of $\operatorname {...