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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Normal extensions problem in Lang

This is a problem in Lang's Algebra. $F$ is finite normal extension over $k$ and $f(x)$ is irreducible in $k[x]$. If $f(x)=g(x)h(x)k(x) \in F[x]$ where $g(x),h(x)$ are monic irreducible factors in ...
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1answer
54 views

What are the $2$-dimensional algebras over any arbitrary field?

As a follow-up of this question, I would like to ask, what are the $2$-dimensional algebras over $\mathbb R$, $\mathbb Q$, or any arbitrary field? Can we classify them?
3
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1answer
64 views

If the degree of $\alpha=5555$, what is the degree of $\alpha^2$?

$\alpha$ is an algebraic number with degree $5555$. What is the degree of $\alpha^2$? Here are my thoughts so far: I think it is true that $\mathbb{Q} \subseteq \mathbb{Q}[\alpha^2] \subseteq ...
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1answer
79 views

On Transcendence Degrees

Let $A$ and $B$ be integral domains with $A\subset B$. The transcendence degree of $B$ over $A$ is defined as the transcendence degree of $Quot(B)$ over $Quot(A)$. Denote it by $trdeg(B/A)$. Let $S$ ...
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2answers
115 views

For a subfield $K$ of $\Bbb C$ with $K\nsubseteq \Bbb R$, show $K$ is dense in $\Bbb C$. [duplicate]

Let $K $ be a subfield of $\mathbb C$ not contained in $\mathbb R$. Is $K$ dense in $\mathbb C$? My problem is I have never used the concept of dense set in algebra and neither have any idea ...
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0answers
60 views

P. Morandi on p-closure of a field

I am stuck on a step of the proof of Lemma 18.4 of Patrick Morandi, Field and Galois Theory. Let $p$ be a prime number and let $F$ be a field with $\mbox{char}(F) \neq p$. Morandi defines the ...
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1answer
41 views

Construction of a finite field

Let $Z[X]$ denote the ring of polynomials in $X$ with integer coefficients .Find an ideal $I$ in $Z[X]$ such that $Z[X]/I$ is a field of order $4$. My attempt:I know that if $F$ is a field & ...
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1answer
56 views

A field with four elements

Determine the additive group of the field of four elements. My attempt:Consider $(F,+,.) $ the field of four elements.Now $0,1\in F$ as $(F,+,.) $ is a field .As it contains $4$ elements $\exists ...
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1answer
139 views

Algebraic closed field has infinite many elements

Show that an algebraic closed field must have infinite many elements. Let's suppose that an algebraic closed field $K$ has finite many elements. But how could I get a contradiction??
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1answer
60 views

Show that $\mathbb{Z}_3(a)$ is a splitting field of the polynomial

We consider the polynomial $f(x)=x^2+1 \in \mathbb{Z}_3[x]$. We symbolize as $a$ a root of $f(x)$ in an algebraic closure $\overline{\mathbb{Z}}_3$ of $\mathbb{Z}_3$. Show that $\mathbb{Z}_3(a)$ is ...
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1answer
73 views

Show that $f(x)=g(x^{p^e})$ and each root has the same multiplicity

Let $K$ a field of characteristic $p\neq 0$ and let $f(x) \in K[x]$ be an irreducible polynomial. Show that $f(x)=g(x^{p^e}), e \geq 0$, where $g(x)$ is an irreducible separable polynomial of ...
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1answer
45 views

Irreducible polynomial over $\mathbb F_5$

In my book text of Galois theory: The polynomial $X^5-X-1$ over $\mathbb Z$. In $\mathbb F_5$ is irreducible. Could you explain why? Thanks.
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1answer
24 views

Let $G/H$ be a Galois extension and let $N_1$ and $N_2$ be subfields between $G$ and $H$

Show that: $Gal(G/(N_1N_2)) = Gal(G/N_1) \cap Gal(G/N_2)$ Um, one direction seems pretty obvious by definition: I believe it's that $Gal(G/N_1) \cap Gal(G/N_2) \subseteq Gal(G/N_1N_2)$ Now I have ...
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1answer
92 views

Field with 729 elements.

Let $\mathbb{F}$ be a field with 729 elements. How many distinct proper subfields does $\mathbb{F}$ contain. Please be generous and tell the reason also. Thanks.
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1answer
126 views

Problem on a subfield being dense in $\mathbb C$ [duplicate]

Let $K$ be a subfield of $\mathbb C$ not contained in $\mathbb R$. Is $K$ always dense in $\mathbb C$? I have studied to show a set $A$ is dense in $B$ we will have to show that for any element ...
5
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1answer
113 views

Find the degree $[E:\mathbb{Q}]$

Let $p$ a prime number. Find a splitting field $E$ of the polynomial $x^p-2 \in \mathbb{Q}[x]$. I have done the following: The solutions of $x^p-2=0$ are : $$\sqrt[p]{2}, \sqrt[p]{2}\omega, \dots, ...
0
votes
1answer
66 views

$E$ is a splitting field of $f(x)$

Let $f(x)=x^2-2 \in \mathbb{Z}_5[x]$. $f(x)$ is irreducible. Let $\xi$ be a solution of $f(x)$ in an extension of $\mathbb{Z}_5$. How can I show that $E=\mathbb{Z}_5(\xi)$ is a splitting field of ...
1
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1answer
197 views

Examples of a complete ordered field

We know that every complete ordered field is isomorphic to $\mathbb R$, but are there examples of complete ordered fields different, not isomorphically different of course, from $\mathbb R$?
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1answer
95 views

Galois extensions questions

I'm working on answering this question but I'm unsure about alot the way I'm going about with the answers. The question is: Let L be a subfield of $\mathbb{C}$. a) Show that $\mathbb{Q}\subseteq ...
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1answer
94 views

The exponential extension of $\mathbb{Q}$ is a proper subset of $\mathbb{C}$?

This question come from a recent post Exponential extension of $\mathbb{Q}$. An exponential field is a field $\mathbb{K}$ where it's well defined a function $E:\mathbb{K} \rightarrow \mathbb{K}$ ...
2
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1answer
44 views

Model theory of valued

I am currently reading these notes on model theory of valued fields, in the section 3.3 appears this theorem: Theorem. Let $K$ and $L$ be valued fields, with residue fields $k_K$ and $k_L$ ...
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0answers
25 views

Is $PE$ is purely inseparable over $E$?

I want to prove the statement : if $K\le E\le F$ and $F$ is separable over $E$, then $P\subset E$. Here, $P=\{u\in F:\text{$u$ is purely inseparable over $K$}\}\le F$. I claim that $PE$ is both ...
0
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2answers
323 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
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1answer
43 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
3answers
252 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
695 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 ...
9
votes
1answer
396 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
170 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 ...
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1answer
194 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
48 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
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0answers
55 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 ...
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2answers
986 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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1answer
29 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 ...
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1answer
85 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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1answer
82 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
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2answers
69 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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2answers
78 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 $\mathbb{Q}[x,y]/(x,y)$ ...
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2answers
104 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
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0answers
90 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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1answer
45 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 ...
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3answers
139 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 ...
0
votes
2answers
146 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 ...
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2answers
318 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?
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2answers
224 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
1answer
43 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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1answer
34 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?
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2answers
140 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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0answers
38 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
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1answer
84 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 $\operatorname{Aut}_{K}{K(x)}$ ...
2
votes
2answers
39 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 ...