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 ...

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2answers
85 views

Polynomial rings- multiplicative inverse

I need to solve the following question in ring theory. Show that $(Q[x])/\langle{x^2+x+4}\rangle$ is a field. To show that $(Q[x])/\langle{x^2+x+4}\rangle$ is a field, the only thing I need to do ...
3
votes
1answer
17 views

Transcendental extension - Intermediate fields

A friend and I have been trying to prove the following: Let $K$ and $E$ be two fields, and let $u$ be transcendental over $K$. If $K\subset E\subseteq K(u)$, then $u$ is algebraic over $E$. There's ...
3
votes
2answers
58 views

Embedding of a field in a cyclic extension

Show that $K=\mathbb{Q}(\sqrt {a})$ for $a\in \mathbb{Z}$, $a<0$ can not be embedded in a cyclic extension whose degree over $\mathbb{Q}$ divisible by 4. I have tried for order exactly ...
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2answers
75 views

Prove $-(-a) = a$

Let $F$ be a field and $a \in F$. Prove $-(-a) = a$. So we want to show that $(-a) + (-(-a)) = 0$, since inverses are unique (I successfully proved that inverses are unique in an earlier problem ...
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votes
0answers
142 views

What is the overall idea of Galois theory?

I am a third year undergraduate, doing a course on Field and Galois theory. Now, while I seem to understand most of the concepts locally, I do not seem to get the 'Whole picture' of what is happening ...
3
votes
4answers
102 views

Norm of an element in cyclotomic extension (Exercises VI.19 Lang's Algebra)

Let $\zeta$ be a primitive $n^{\rm{th}}$ root of unity. Let $K=\mathbb{Q}(\zeta)$. If $n=p^r (r\geq 1)$ is a prime power, show that $N_{K/F}(1-\zeta)=p$ If $n$ is divisible by at least two distinct ...
4
votes
1answer
42 views

Equivalent conditions of a Galois extension (Exercise VI.4 in Lang's Algebra)

let $k$ be a field of characteristic $\neq 2$. Let $c\in k, c\notin k^2$. Let $F=k(\sqrt{c})$ . Let $\alpha=a+b\sqrt{c}$ with $a,b\in k$ not both $a,b=0$. Let $E=F(\sqrt{\alpha})$. Prove that the ...
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1answer
47 views

Does $K(x)=E(x)$ imply $K=E$?

Let us suppose we have two fields $K$ and $E$ and $K\subseteq E$. Is it true that $K(x)=E(x)$ implies $K=E$? I know it seems sort of obvious, but I don't know if it is actually true. It is for a step ...
2
votes
0answers
43 views

Is the extension normal and a little work check.

Let us consider the polynomial $f(x)=x^3+x^2-4x+1$. I was asked the following things: (1) Prove that $f(x)$ has one and only one negative root. For this I just used Bolzano's theorem and noticed it ...
3
votes
2answers
441 views

Why is 1+1=0 in a finite field F={0,1}?

This table: $$\begin{array}{|c|cc|} \hline +& 0& 1\\ \hline 0& 0& 1\\ 1& 1& 0\\ \hline \end{array}$$ "feels" right, but how can you prove that $1+1=0$? What is the reason? I ...
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votes
1answer
26 views

Confusion about coordinate function

Let $F_{27}=\{0, \alpha, \dots, \alpha^{26}=1\}$ and $B=\{1, \alpha, \alpha^2\}$ be a basis of $F_{27}$ over $F_3$ then an element $\alpha^k = c_1+c_2\alpha+c_3\alpha^2$ where $1\le k\le 26$. To ...
1
vote
1answer
19 views

normality of algebraic closure

is it always true that $F'|F$ is a normal extension?$F'$ means the algebraic closure of $F$. what conditions are necessary for that? thanks
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vote
1answer
40 views

Smallest subring containing $\sqrt{5}$

I want to find the smallest subring of $\mathbb R$ which contains $\mathbb Q$ and $\sqrt 5$. I am sure that$\{a+b\sqrt{5}:a,b \in \mathbb Q \}$ is the right candidate. I already showed that this is ...
0
votes
0answers
11 views

The notation $EF$ (the field generated by $E$ and $F$)

I came across this notation. Please read the question : Show that if $E$ and $F$ are normal extensions of $K$ within a field $U$, then $EF$, the subfield generated by $E$ and $F$, and $E\bigcap F$ ...
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votes
0answers
30 views

Proof number fields

In the proof Theorem 5.2 page 22 Janusz Number fields, that says Therem 5.2: The finite dimensional field extension $L$ of $K$ is separable if and only if the bilinear form $(x,y)=T_{L/K}(xy)$ from ...
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2answers
46 views

A problem about normal extensions and automorphisms

this is my problem: Suppose $K|F$ is a normal extension. Prove that for every $\alpha ,\beta \in K$ that have the same minimal polynomial over $F$ there is a $F$-algebra automorphism of $K$ ...
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vote
2answers
37 views

Maximal ideal in $\mathbb{Q}[x,y]$

I am trying to prove that $(x,y)$ is a maximal ideal of $\mathbb{Q}[x,y]$. Since an ideal $I \subseteq R$ is maximal if and only if $R/I$ is a field, it suffices to prove that $\mathbb{Q}[x,y]/(x,y)$ ...
0
votes
2answers
73 views

A problem about splitting field and irreducibility of a polynomial

Suppose that $K$ is the splitting field of $f(x)\in F[x]$, when the degree of $f(x)$ is $n$ and $[K:F]=n!$. Show that $f(x)$ is irreducible over $F$. i know that $K|F$ is normal,but i don't know ...
3
votes
0answers
58 views

Splitting field in finite field

What is the splitting field of the polynomial $X^{p^8}-1$ over $\mathbf F_p$? I'm confused, is not $X^{p^8}-1=(X-1)^{p^8}$ then the splitting field is $\mathbf F_p$? Thanks.
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votes
1answer
43 views

Suppose you are given an ordered field $F$. You dont know exactly what set $F$ is, but…

Suppose you are given an ordered field $F$. You dont know exactly what set $F$ is, but you know there exists a nonempty subset $A\subset F$ with no upper bound. What can we say about $F$? Namely, can ...
3
votes
3answers
87 views

Proving two finite fields are isomorphic

So I'm asked to prove that $\mathbb{F}_9$, defined as $\{ a+bi$ | $a,b \in \mathbb{Z}_3,$ $i^2 = 2 \}$, is isomorphic to the field $F_1$, defined as $\mathbb{Z}_3[x]/ \langle x^2+2x+2 \rangle$, where ...
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0answers
10 views

Can cancellation property of finite vector space be used to show uniqueness of vectors combination?

Here are several questions together, but they need to be related (hopefully). If set of vectors is a basis, then, considered with all its linear combinations it forms a field (with vector addition ...
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2answers
46 views

Does every infinite field contain the integers as a subring?

I simply ask because if $1+1=2(1)=2$ then this would imply that all positive integers are contained, and as every element in a field has a negative all the negative integers are contained. At the same ...
0
votes
0answers
21 views

Basis for a field extension

Suppose that $K=\mathbb Q(\alpha)$ is normal and $\alpha,\alpha_2,\ldots,\alpha_n$ the conjugates of $\alpha$. Is necessarily $\{\alpha,\alpha_2,\ldots,\alpha_n\}$ a basis for $K$ over $\mathbb Q$? ...
2
votes
2answers
68 views

How to find a minimal polynomial

I need to find minimal polynomial of $\alpha = \sqrt 2 + \sqrt [3] 3 $ over $\mathbb Q$ and prove that my result is minimal polynomial. How do I do that?
2
votes
2answers
29 views

Example of a normal extension.

Can you give an example of a Normal extension which is not a splitting field of some polynomial.? I know that splitting field of a polynomial is always a normal extension but i am looking for the ...
0
votes
0answers
18 views

Norm in cyclotomic field

Suppose $p$ is a rational prime and $\zeta=e^{2\pi i/ p}$. Prove that the groupp of non-zero elements of $\mathbb Z_p$ is cyclic, show that there exists a monomorphism $\sigma:\mathbb Q(\zeta)\to ...
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vote
1answer
39 views

Proof about field extension : A geometric way

Let $M \subset \mathbb C $ be a sub-field which is not contained in $\mathbb R$ and which is closed under complex-conjugation. Let $L(M)$ be the set of all lines which crosses two points of $M$ and ...
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votes
1answer
24 views

Transcendental extension over a field K.

Prove that $x$ is transcendental over $F(x)$ or more generally show that any element not in $K$ but in $K(x_1,x_2,x_3,x_4,\ldots,x_n)$ is transcendental?
2
votes
2answers
45 views

Extension field, degree of $[\mathbb Q(i,\sqrt{-3}):\mathbb Q]$

I want to calculate the degree of $[\mathbb Q(i,\sqrt{-3}):\mathbb Q]$, can I do like that: $$X=i+\sqrt{-3}\implies X=i(1+\sqrt{3})\implies X^2=-(1+\sqrt{3})^2\implies X^2=-1-2\sqrt{3}-3\implies ...
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votes
0answers
27 views

Roots Of An Inseparable Polynomial.

Suppose that we have a field $K = \mathbb{Z}_p(t)$ where $p$ is prime. (So this is the field of rational polynomials with $t$ as the variable) Let $f = x^p - t$ be a polynomial in $K[x]$. How can ...
2
votes
1answer
43 views

Fixed field of two subgroups of $\operatorname{Aut}_{K}{K(x)}$

This link explains $\operatorname{Aut}_{K}{K(x)}$. And I want to know how to solve two problems below in the Hungerford's Algebra, p.256. $7.$ Let $G$ be the subset of $\operatorname{Aut}_{K}{K(x)}$ ...
1
vote
2answers
25 views

Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb Q[\alpha]|)=1$

Please, help me to understand this problem: Let $\alpha=\sqrt[3]{2}$ be a root of the polynomial $x^3-2$. a) Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb ...
1
vote
1answer
39 views

linear independence and field extension

suppose $K|F$ is a field extension & $\alpha \in K$ is such that $[F(\alpha):F]>=n$, if $\lambda_1,...,\lambda_n$ are distinct scalars of $F$,prove that ...
1
vote
1answer
42 views

Suppose that $L:K$ is algebraic. Show that the following are equivalent:

$(A)$ $L:K$ is normal $(B)$ if $j$ is any monomorphism from $L$ to $\overline{L}$ which fixes $K$ then $j(L) \subseteq L$ $(C)$ if $j$ is any monomorphism from $L$ to $\overline{L}$ which fixes $K$ ...
3
votes
1answer
31 views

The condition that $[LM:K]=[L:K][M:K]$ holds in the field.

Let $L$ and $M$ be intermediate fields of the extension $K\subset F$, of finite dimension over $K$. Assume that $L\cap M=K$ and $[L:K]$ or $[M:K]$ is 2, then $[LM:K]=[L:K][M:K]$ holds. How can I use ...
4
votes
1answer
35 views

Suppose $F$ is a finite field of characteristic $p$. Prove $\exists u \in F$ such that $F = \mathbb{F}_{p}(u)$.

Suppose $F$ is a finite field of characteristic $p$ ($p$ a prime). Prove $\exists u \in F$ such that $F = \mathbb{F}_{p}(u)$. Here, $\mathbb{F}_{p}$ denotes the field with $p$ elements. Here is ...
0
votes
1answer
37 views

field of fractions and being algebraically closed

prove that for every field $F$ the field of fractions $F(x)$ is not algebraically closed. it is a problem which i don't know how to deal with it. help please. thank you.
1
vote
1answer
59 views

$f(x)$ is still irreducible

Let $f(x) \in K[x]$ an irreducible polynomial of $K[x]$ of degree $n$. Let $K\leq F$ a field extension with $[F:K]=m$. If $(n,m)=1$ show that $f(x)$ stays irreducible also as a polynomial of ...
2
votes
2answers
27 views

monomorphism on an algebraic field extension

let $E|K$ be an algebraic extension and $\phi:E\rightarrow E$ a $K $-algebra monomorphism,prove that $\phi$ is onto. i assume $\alpha\in E-\phi(E)$ to make a contradiction and i assume $f(x)$ to be ...
1
vote
0answers
42 views

Diagrams characterizing ring characteristic, and in particular field characteristic 1?

For a given prime $p$, what is the diagram for saying (e.g. in some category where morphisms are field homomorphisms) that a field has characteristic $p$? Is there a diagram, or the shape of what ...
0
votes
2answers
9 views

Does $\sum_{i=1}^nx_i=0$ and $\sum_{i=1}^nc_ix_i=0$ implies $c_i=c_j$ in a field?

Does $\sum_{i=1}^nx_i=0$ and $\sum_{i=1}^nc_ix_i=0$ implies $c_1=\cdots=c_n$ where $c_i$ and $x_i$ are elements of a field $F$? If so, why?
1
vote
3answers
43 views

What are all different (non-isomorphic) field structures on $\mathbb R \times \mathbb R$

We know that $\mathbb R \times \mathbb R$ forms a field under addition and multiplication defined as $(a,b)+(c,d)=(a+c,b+d)$ ; $(a,b)*(c,d)=(ac-bd,ad+bc)$ ; is there any other way to make $\mathbb R ...
2
votes
1answer
29 views

Show that it is a subfield

To show that $$\mathbb{Q}(2^{1/x}, 2^{1/y}) \subseteq \mathbb{Q}(2^{1/{xy}})$$ knowing that $(x,y)=1$, $x, y \in \mathbb{N}$ can we do the following?? $$2^{\frac{1}{x}}=2^{\frac{y}{xy}}=\left ( ...
0
votes
1answer
21 views

Problem with modulo in field

I have problem with comprehending how works number in field when it's rasied to negative power. For instance if we have $4^{-1}$ at $Z_{5}$ I tried to write it as $4\cdot 4^{-1}+4^{-1}=4^{-1}(1+4)$ ...
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votes
0answers
19 views

Why does this criterion imply that $A$ is a subfield of $E$?

$E$ is an extension field of a field $F$ and $A$ is the subset of $E$ containing all the members algebraic over $F$. "To prove that $A$ is a subfield of $E$ it is enough to show that any two elements ...
1
vote
0answers
29 views

Algebraic extension of rational functions

Let $k\subset F\subseteq k(X)$ be chains of field extension, prove that $k(X)/F$ is algebraic. "Proof:" Let $y\in F\setminus k$ then $y=\frac{P(X)}{Q(X)}$ with $P\notin k$ or $Q\notin k$. It ...
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votes
0answers
8 views

A question from calculus my test(Curl, guess theorem )

the value of the integral $$ \iint rotF*n*ds \quad where \quad s-> x^2+y^2+z^2=4 \quad $$ and the normal is making a blunt angle with the Z axis, and $$ f=(zsinx-2y+1)i+(3x)j+(4xz+z^3)k $$ im ...
0
votes
2answers
45 views

Subgroups of $\mathbb F_{p^n}$

Is it possible to give a discription of the possible subgroups (with respect to $+$) of the finite field $\mathbb F_{p^n}$ (obviously, $p$ is a prime number). Of course, if $n = 1$, $(\mathbb F_p,+)$ ...
3
votes
1answer
24 views

different factors of irreducible polynomials over a Galois extension does not share roots

Let $F$ be a field and $E/F$ a galois extension. Let $f\in F[x]$ be an irreducible polynomial. I'm trying to prove that all the irreducible factors of $f(x)$ over $E[x]$ have the same degree. If ...