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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1answer
21 views

quadratic equation modulo some number

I read a post that $$ax^2+bx+c \equiv 1 \pmod p$$ can be solved in a similar way we solve a simple quadratic equation, just by replacing division by $2a$ by modulo inverse of $2a$ and square root of ...
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2answers
28 views

Diagonalisable matrices over different fields

I believe this fits in with my knowledge of Jordan Normal form, however I am not sure how to approach the question itself? I am especially lost with $F_7$
4
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2answers
327 views

How to transform a general higher degree five or higher equation to normal form?

This is a continuation of How to solve fifth-degree equations by elliptic functions? How to transform a general higher degree five or higher equation to normal form? For xeample, a quintic equation ...
-1
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3answers
41 views

As fields $F$ not isomorphic with $\mathbb R$ , but as sets $F \sim \mathbb R$ , example ? [closed]

I am looking for an example of a field $(F,+,.)$ such that as fields , $F$ , $\mathbb R$ are not isomorphic but there exist a bijection between $F$ and $\mathbb R$ .
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1answer
45 views

Example of infinite field $(F,+,.)$ such that $(F^*, . )$ is a cyclic group ? [duplicate]

It is known that any infinite cyclic group can never be a vector space , from this we can derive that if $(F,+,.)$ is an infinite field then $(F,+)$ cannot be cyclic . I am asking , is there any ...
1
vote
1answer
28 views

Normal field extension implies splitting field

I feel like this fact should be easy but I'm struggling to see it. If I have a polynomial $f \in K[x]$ which is irreducible and has roots $\alpha$, $\beta$ in some finite normal (over $K$) extension ...
1
vote
1answer
31 views

Calculating fixed fields

I have to show that $\mathbb{Q}(\zeta)^{<\sigma>}$ $=$ $\mathbb{Q}(\zeta + \frac{1}{\zeta})$ $\sigma \colon L \to L$ is defined by $\sigma(\alpha) = \overline{\alpha}$, where ...
2
votes
3answers
257 views

Upper bound on cardinality of a field

Is there an upper bound on the cardinality of a field? The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than ...
2
votes
1answer
36 views

Algebraic invariants for first order equivalence between fields

I know that every two models of the theory $ACF$ (namely two algebraic closed fields) with the same characteristic are elementary equivalent. But what about generic fields? Are there any algebraic ...
2
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0answers
35 views

The number of bijective polynomials of particular degree in a field

I need to know please: In a finite field of q elements how many bijective polynomials exist whose degree are smaller than d?
1
vote
1answer
43 views

Degree 4 extension of $\mathbb {Q}$ with no intermediate field

I am looking for a degree $4$ extension of $\mathbb {Q}$ with no intermediate field. I know such extension is not Galois (equivalently not normal). So I was trying to adjoin a root of an irreducible ...
0
votes
0answers
17 views

Galois group is $S_{n}$

If $K/F$ is Galois extension with Galois group $S_{n}$ then show that $K$ is the splitting field of a degree $n$ polynomial irreducible over $F$. We know $K$ is splitting field of some separable ...
1
vote
1answer
35 views

Can every element of an algebraic field extension, $E \subseteq F$ be represented as $f(s)$, where $s$ is another member and $f(x) \in F[x]$

Specifically, could every member of $E$ be generated by one element $s$, by evaluating different polynomials of $s$ with coefficients in $F$?
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votes
2answers
24 views

Field extensions and minimal polynomial

I have to find the degree of $1+\sqrt[3]{2}+\sqrt[3]{4}$ over $\mathbb{Q}$. This is what I found already: $\mathbb{Q}(1+\sqrt[3]{2}+\sqrt[3]{4})=\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{4})$ ...
0
votes
1answer
24 views

Order of automorphism = Order of field extension property

I've read that if $L/K$ is a field extension and $|Aut(L/K)|=|L/K|$, then L/K is a galois extension. I was wondering whether the converse is true, i.e if $|Aut(L/K)|\neq |L/K|$, then can we just ...
0
votes
1answer
24 views

Is Q(4th root(2))/Q a galois extension

I'm having some difficulty with the definition for galois extensions. The definition as read from my notes is $L/K$ is galois if $L^{Aut(L/K)}= K$. Where $Aut(L/K)$ is definied to be the set of all ...
1
vote
1answer
44 views

Degree of splitting field less than n!

I've been asked to prove that if a function $f\in \mathbb{Q}$ has degree $n$, then the splitting field of $f$ has degree less than or equal to $n!$. That is $\mathbb{Q}(\alpha_1, ...
1
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0answers
40 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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0answers
40 views

Definition of $K[x,y]/\langle f(x,y), g(x,y)\rangle$

Could you explain me the definition of $$K[x,y]/\langle f(x,y), g(x,y)\rangle?$$ How can we show for example that $$\mathbb{C}[x,y]/\langle x-1, y+x^2-1\rangle $$ is a field?
1
vote
1answer
62 views

Can someone explain me the sentence about ideals?

Can someone explain me the sentence: "If $R=K[x]$ the prime ideals are $\langle f(x)\rangle $ where $f(x)$ is an irreducible polynomial in $K[x]$ and $\langle 0\rangle $, and again $\langle ...
2
votes
1answer
45 views

Find a non-principal ideal (if there exists any) in the rings Z[x], Q[x], Q[x, y]

I know that $Q$ is a field, which makes $Q[x]$ a PID, which means there are none. I'm having trouble with the notation for ideal generators, and i know the $Z[x]$ has to do with something that looks ...
6
votes
1answer
72 views

The absolute Galois group of a finite field is strongly complete

Let $k$ be a finite field. I am trying to prove that the absolute Galois group of $k$, i.e., $G = \operatorname{Gal}(\bar{k} / k)$ where $\bar{k}$ is an algebraic closure of $k$, is strongly ...
4
votes
1answer
53 views

Can a field be isomorphic to its subfield but not to a subfield in between?

A question related to this one Can a field be isomorphic to its subfield?: are there field extensions K/E and E/F such that K and F are isomorphic but E is not isomorphic to them?
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2answers
42 views

Show that there are no $f_1, f_2$ such that $f_1f_2 = f$.

Let $\mathbb{F}$ a finite field. Show that there's $f\in\mathbb{F}[x]$ such that $\deg(f)=2$ and there are no linear polynomials $f_1,f_2\in \mathbb{F}[x]$ such that $f_1f_2 = f$. Hint: Define ...
1
vote
3answers
61 views

Do we know if all simple extensions of the field of rational numbers by transcendental numbers are not equal?

I understand that $\mathbb{Q}(x) \cong \mathbb{Q}(u)$ for all transcendental $u$, where $\mathbb{Q}(x)$ is the field of rational forms over $\mathbb{Q}$ and thus that all simple extensions of the ...
0
votes
1answer
46 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?
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0answers
11 views

Why $f(0)\neq 0$ where $f $ is a polynomial over the field $F_q$ and $deg(f)=m > 0$?

To construct the Residue class ring $F_q[x]/(f)$ having $q^m-1$ non-zero elements. Is it necessary for $f(0) \neq 0$? Why or why not? I have worked with different examples such as $x^3+x=f \in ...
4
votes
3answers
97 views

Show that $\sqrt [3]{2}-\sqrt [3]{4}$ is algebraic

How do I show, step by step, that $\sqrt [3]{2}-\sqrt [3]{4}$ is a root of $x^3+6x+2$? Start with $x=\sqrt [3]{2}-\sqrt [3]{4}$ do not use the cubic, the cubic is given for convenience. ( This is ...
2
votes
0answers
18 views

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 ...
3
votes
1answer
53 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 ...
13
votes
3answers
555 views

Characterization of a subfield $K \varsubsetneq \mathbb {C}$ and $x\in \mathbb{R}$

Characterize $x \in \mathbb R$ such that there exist a subfield $K \varsubsetneq \mathbb C$ such that $K(x) = \mathbb C$. -All subfields $K$ of $\mathbb{C}$ contain $\mathbb Q$, then all $x\notin ...
5
votes
4answers
241 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.
0
votes
2answers
77 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 ...
0
votes
1answer
43 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 ...
4
votes
1answer
39 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 ...
0
votes
0answers
29 views

The number of Permutation polynomial in a field

I need to know,please: (1) How many permutation polynomial exist in a finite field (any field)? (2) Is there any way to pick a random permutation polynomial in this field?
0
votes
1answer
32 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 & ...
0
votes
1answer
31 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 ...
1
vote
1answer
25 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.
0
votes
1answer
23 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 ...
0
votes
1answer
60 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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2answers
42 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 $K$ ...
3
votes
4answers
98 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 ...
2
votes
1answer
52 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
38 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
vote
1answer
71 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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0answers
37 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 $\mathbb R$?
2
votes
1answer
58 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 $E:\mathbb{K} \rightarrow \mathbb{K}$ ...
0
votes
2answers
49 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 ...
2
votes
1answer
33 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 $k_L$ ...