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
38 views

Reducing splitting field

If we have splitting field: $$L=\mathbb{Q}(\sqrt{2+i\sqrt{6}},\sqrt{2-i\sqrt{6}}) $$ we can multiply these two zeroes and get $\sqrt{10}$ so we have $$L=\mathbb{Q}(\sqrt{2+i\sqrt{6}},\sqrt{10})$$ ...
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2answers
47 views

Are the fields $\mathbb{Q}(\sqrt[7]{16}+3 \sqrt[7]{8})$ and $\mathbb{Q}(\sqrt[7]{16})$ equal?

I have trouble with these field extensions. Is field $\mathbb{Q}(\sqrt[7]{16}+3 \sqrt[7]{8})$ equal to field $\mathbb{Q}(\sqrt[7]{16})$? We can $\sqrt[7]{16}+3 \sqrt[7]{8}$ express as ...
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1answer
31 views

Cyclic extension without primitive root of unity

Let $F$ be a field that doesn't contain a primitive fourth root of unity. Let $L = F(\sqrt a)$ for some $a \in F - F^2$ and let $K=L(\sqrt b)$ for some $b \in L - L^2$. If we have $N_{L/F}(b) ...
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3answers
42 views

List the elements of the field $K = \mathbb{Z}_2[x]/f(x)$ where $f(x)=x^5+x^4+1$ and is irreducible

Since $\dim_{\mathbb{Z}_2} K = \deg f(x)=5$, $K$ has $2^5=32$ elements. So constructing the field $K$, I get: \begin{array}{|c|c|c|} \hline \text{polynomial} & \text{power of $x$} & ...
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1answer
37 views

Unique isomorphisms and universal properties

Having not studied category theory, I'm trying to piece together without using category theory what is meant when an algebraic structure is said to possess a universal property or be unique up to ...
3
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2answers
58 views

Determine which of the following rings are fields.

Have I done it correctly? Determine which of the following rings are fields: a) $(\mathbb{Z}/2\mathbb{Z})[x]$/$\large_{(x^2+1)}$ b)$(\mathbb{Z}/3\mathbb{Z})[x]$/$\large_{(x^2+1)}$ My ...
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1answer
64 views

What are the groups $\text{Hom}(F^\times\!, F^+\!)$ and $\text{Hom}(F^+\!, F^\times\!)$?

Background. Exercise 36 in Rose's A Course On Group Theory reads Prove that there is no field $F$ with $F^\times \cong F^+$. The problem is solved in characteristic $\ne 2$ by considering $-1$ ...
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1answer
57 views

Proof that every field $F$ has an algebraic closure $\bar F$

I am reading the book A First Course in Abstract Algebra written by Fraleigh and I do not really understand the proof of theorem 31.22, that every field $F$ has and algebraic closure $\bar F$. I ...
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0answers
38 views

How to solve the equation $x^2+Dy^2=\alpha$ over finite fields

It is known that the equation $x^2+Dy^2=1$ is solved over finite fields $\mathbb{F}_q$ and we can point out the solutions . I wonder can we give solutions for the equation $x^2+Dy^2=\alpha$ for any ...
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1answer
43 views

Question from 14.6 “Galois Groups of Polynomials” from Dummit and Foote

I am confused in the proof of proposition 30 in Dummit and Foote on page 608. Near the end of this "proof" he goes on to say, By the Fundamental Theorem of Galois Theory, the fixed field of ...
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2answers
56 views

How can we work with field extensions when our base fields aren't actually subfields?

I've been wondering this for a little while. Say we are working with the rational numbers $\mathbb{Q}$, and then we wish to talk about the extension fields $E$ of $\mathbb{Q}$, by which we mean the ...
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0answers
31 views

Galois group for the algebraic closure of a finite field

If $N$ is the algebraic closure of a finite field $F$, prove that $\operatorname{Gal}(N/F)$ is an abelian group and that any element of the Galois group has infinite order. If $N$ were a finite ...
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1answer
12 views

Splitting of primes terminology doubt

What do we mean when we say that a given prime $p$ splits completely in an algebraic extension of $\mathbb Q$? Are we talking about the splitting of prime ideals into unique factors? And, in that ...
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2answers
27 views

How should I find Splitting Field of $x^3-2$ over $\mathbb Q$.

How should I find Splitting Field of $x^3-2$ over $\mathbb Q$. **My try **: $x^3-2=(x-2^\frac{1}{3})(x^2+2^\frac{1}{3}x+2^\frac{2}{3})$ On solving I am getting the roots as ...
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1answer
38 views

Field Question Proofs

True or False: In every field $F$, if $x,y$ belong to $F$ and $w,w'$ belong to $F$ such that $x * w = 1$ and $y * w' = 1$, then $(x * y) * (w * w') = 1$. I think the answer would be false mainly ...
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0answers
21 views

Galois closure uniqueness confusion

I'm a little bit confused by the statement of this corollary, (Corollary 23 pg 594 of Dummit and Foote) Let $E/F$ be any finite separable extension. Then $E$ is contained in an extension $K$ which is ...
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1answer
49 views

From where can I study more about Dickson polynomials?

I know some basic bits about this construction as to how they effect permutations of Galois fields. But I want to get some detailed understanding of them. Any references?
2
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1answer
29 views

Field extension of prime degree

Question: Let $L$ be the extension of the field $K$ such that $[L:K]=p$, where $p$ is a prime number, and $\alpha \in L$. Prove that $K(\alpha)=K$ or $K(\alpha)=L.$ Proof: From $$ \alpha \in L ...
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0answers
32 views

Find and show primitive element of field extension

I have the polynomial $\ f(x)= x^{15}-3 $ and want to find the splitting field. The splitting field is surely $\Bbb Q$ adjoined the roots of the polynomial. I want to write the splitting field as ...
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0answers
38 views

What does this theorem mean, exactly?

The polynomial $X^{p^n}-X$ is precisely the product of all the distinct irreducible polynomials in $F_p[X]$ of degree $d$ where $d$ runs through all the divisors of $n$. I don't even get the ...
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0answers
8 views

Question on separable degree

I asked a similar question before but I didn't get a satifying answer, so I'm posting it again. Let me first define terms: Def1 Let $E/F$ be an algebraic field extension and $\bar F$ be an ...
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1answer
76 views

If there are $k_1, k_2 \in K$ such that $K(\alpha + k_1\beta)=K(\alpha + k_2\beta)$ then $K(\alpha,\beta) = K(\alpha + c\beta)$ for some $c \in K$.

Here is the problem: "Let $K \subset M$ be a finite field extension, and $\alpha, \beta \in M$. Suppose there are $k_1, k_2 \in K$ are distinct and such that $K(\alpha + k_1\beta)=K(\alpha + ...
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0answers
34 views

What is the possible number (supremum) of subfields of $\mathbb{F}$?

Let $\mathbb{F}$ be field. it is a finite dimensional extension over $\mathbb{Q}$. So let $B=\{v_1, v_2, v_3, ... , v_n\}$ be a basis for $\mathbb{F}$ over $\mathbb{Q}$. From the finite dimension ...
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1answer
45 views

Cyclotomic field over $\Bbb Q$

Let $K$ be cyclotomic field generated over $\Bbb Q$ by the $9$th root of unity $z$, having Galois group $G$. Show that it is a cyclic extension of degree $6$ of $\Bbb Q$ and by making use of the ...
3
votes
3answers
99 views

$\alpha \in \overline{\mathbb{F}}_q$ satisfying $\alpha^{q+1}+\alpha=-1$

Let $\overline{\mathbb{F}}_q$ be the algebraic closure of $\mathbb{F}_q$. Assume that $\alpha \in \overline{\mathbb{F}}_q$ satisfies at $$\alpha^{q+1}+\alpha=-1$$ Show that $\alpha \in ...
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1answer
25 views

Finding proper subfields

Let $\omega$ denote the cube root of unity such that $\omega\neq 1$. I want to find the subfields properly contained in $\mathbb Q(\sqrt[3]{2},\omega)$ and containing $\mathbb Q$ properly. Two of ...
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1answer
33 views

Relation between $\mathbb{Q}(2^{1/3})$ and $\mathbb{Q}(w2^{1/3})$

I have the following exercise in my homework: Are $\mathbb{Q}(2^{1/3})$ and $\mathbb{Q}(w2^{1/3})$ isomorphic, where $w = \textrm{cis}((2\pi)/3)$? Prove your answer. I think they are, but I'm ...
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5answers
70 views

looking for the inverse of $3+2\alpha^2$ over $\Bbb Q(\alpha)$ with $\alpha=\sqrt[3]{2}$

I have $\alpha=\sqrt[3]{2}$ and want to calculate the inverse of $3+2\alpha^2$ over $\Bbb Q(\alpha)$. There's a hint which tells me to look at the minimal polynomial $m_\alpha$ of $\alpha$ over $\Bbb ...
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1answer
31 views

Is the statement of the theorem correct?

I have been asked to prove this:: $f,g$ are polynomials over a field $F$ .Prove that if $f,g$ are relatively prime then $f,g$ have no common roots in any extension of $F$. But I wonder why is ...
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2answers
22 views

Non-algebraic subfield intersection

Construct an example in which a field $F$ is of degree 2 over two distinct subfields $A$ and $B$, but so that $F$ is not algebraic over $A\cap B$. I'm having trouble thinking of an explicit example ...
4
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1answer
47 views

Showing that $\mathbb{Q}(\sqrt{2},\sqrt{3}, \sqrt{(9 - 5\sqrt{3})(2-\sqrt{2})})$ is normal over $\mathbb{Q}$, and finding its Galois group

If $K=\mathbb{Q}(\sqrt{2},\sqrt{3}, u)$, where $u^2 = (9 - 5\sqrt{3})(2-\sqrt{2})$, show that $K/\mathbb{Q}$ is normal, and find $\operatorname{Gal}(K/\mathbb{Q})$. I found that the minimal ...
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1answer
41 views

finding intermediate fields of field extensions: simple method vs. Galois theory

I am looking for a straightforward way to find all intermediate fields of a field extension. Let's take the splitting field of $X^3-2$ over $\Bbb Q$ as an example. If we adjoin the roots of $X^3-2$ ...
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1answer
45 views

How do I find the quotient field of $\mathbb{Z}[\sqrt{d}]$?

Our teacher said sometimes the quotient field is $\mathbb{Q}[\sqrt{d}]$ and sometimes it's $\mathbb{Q}[\frac{1+\sqrt{d}}{2}]$. How do we decide, or what are the conditions on $d$ which helps us to ...
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4answers
35 views

Basis of $\mathbb{Q}[\sqrt[3]{2}]$

How do I prove that $1, \sqrt[3]{2}, (\sqrt[3]{2})^2$ is a basis of $\mathbb{Q}[\sqrt[3]{2}] = \{ a + b \sqrt[3]{2} + c (\sqrt[3]{2})^2\: a,b,c \in \mathbb{Q} \}$. It's one of these cases where the ...
7
votes
1answer
51 views

On the existence of field morphisms

Let $K$ and $L$ be two fields, does the existence of two field morphisms $f\colon K\rightarrow L,\ g\colon L\rightarrow K$ imply that, as abstract fields, $K\cong L$ (not necessarily via $f$ or $g$)?
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1answer
33 views

Exactly one ring homomorphism $F[X] \rightarrow S$

Let $F$ be a field, and $f \in F[X]/(f)$. Let $f$ have a zero point $\alpha$, that is, $f(\alpha)=0$. Let $F$ be a subring of $S$, and $\beta \in S$ with $f(\beta)=0$. Show that there is exactly one ...
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2answers
23 views

Let $K$ be a field and let $p(x)\in K[x]$ be an irreducible polynomial of degree $d$. Let $L = K[x]/p(x)$. Prove that $[L:K] = d$.

I'm not sure where to go with this question. I know that $K[x]/p(x)$ is a field since p$(x)$ is irreducible means it is maximal in $K[x]$.
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3answers
95 views

How do I prove this field homomorphism is an isomorphism?

The question is as follows. Let $F$ be a finite field with unit $1$ not equal to zero. Let the function $f: F \to F$ be given by $f(x) = x^3$, where the $\operatorname{char}(F) = 3$. Prove it is a ...
4
votes
1answer
49 views

Is the primitve element of $\mathbb{Q}[\alpha_1, \alpha_2, \ldots]/\mathbb{Q}$ always $\alpha_1 + \alpha_2 + \cdots$?

I have dealt with a number of algebraic field extensions $\mathbb{Q}[\alpha_1, \alpha_2, \ldots]/\mathbb{Q}$ and the primitive element was always $\alpha_1 + \alpha_2 + \cdots$. Is this generally true ...
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2answers
39 views

Injectivity and norm function on finite fields [closed]

Let $q$ be an odd prime power. Consider the map $f:\Bbb F_{q^3} \rightarrow \Bbb F_{q^3}$, defined by $$f(x)=\alpha x^q+\alpha^q x$$ for some fixed $\alpha \in \Bbb F_{q^3} \setminus \{ 0 ...
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0answers
12 views

Is $K[X]/fK[X]$ = $K[X]/f$?

Let K be field. f is polynomial in K[X]. Is it the same: $K[X]/fK[X]=K[X]/(f)$ ? In other words are the elements equal h+fK[X]=h+f ? ($h \in K[X]$
2
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1answer
14 views

A basis of a field extension $K \subset L$ spans $L(C)$ over $K(C)$ for any subset $C$ of a field $M\supset L$

Let $K \subset L \subset M$ be fields; $\{\beta_1, ..., \beta_k\}$ a basis for $L$ over $K$ and $C$ a subset of $M$. Then $\{\beta_1, ..., \beta_k\}$ generates $L(C)$ over $K(C)$ (where $K(C)$ is the ...
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3answers
85 views

A commutative noetherian ring in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields

PROBLEM A commutative noetherian ring $R$ in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields. I am lost with the condition $I^2=I$ and the desired result "a ...
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0answers
76 views

Extensions of a field?

Prove that there are infinitely many degree 5 extensions of $\mathbb{F}_{121}(x)$. I know that $\mathbb{F}_{121}$ is isomorphic to the splitting field of $x^{121}-x$ over $\mathbb{F}_{11}$, but I'm ...
4
votes
3answers
82 views

What elements may I adjoin to $\mathbb{Q}[\sqrt{3}]$ in order to get to $\mathbb{Q}[\sqrt{7+\sqrt{3}}]$

The field extension $\mathbb{Q}[\sqrt{7+\sqrt{3}}]/\mathbb{Q}$ has degree four and $\sqrt{7+\sqrt{3}}$ is a primitive element. I'm interested in dividing this into two successive field extensions of ...
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0answers
19 views

Roots of a polynomial over a finite field

Let $f(x)=a_0x+a_1x^q+... a_{k-1}x^{q^{k-1}}$ be a nonzero polynomial for a prime $q$. It is easy to observe that $$f:F_{q^n}\to F_{q^n}$$ a linear function. I want to show that $f$ has at most ...
4
votes
1answer
73 views

Finite fields and their subfields

Let $\mathtt{F}$ and $\mathtt{F'}$ be two finite fields of order $q$ and $q'$ respectively. Then: $\mathtt{F'}$ contains a subfield isomorphic to $\mathtt{F}$ if and only if $q\le q'$ ...
0
votes
1answer
32 views

Irreducible polynomial of $\sqrt{2}+\sqrt{7}$ on $\Bbb{Q}$.

I would like to find the irreducible polynomial on $\Bbb{Q}$ of $\sqrt{2}+\sqrt{7}$. How can I do that ? First time I see this kind of question, I can find a polynomial $X^2-2$ witch $\sqrt{2}$ is a ...
1
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0answers
43 views

Theorem with splitting fields

I am trying to understand the following: Theorem I. If the polynomial $p(x)$ is irreducible in $F[x]$ and if $a$ is a root of $p(x),$ then $F(a) \cong F'(b)$ where $b$ is a root of $p'(t) \in ...
3
votes
1answer
24 views

What do the statement mean by "leaves every element of $F$ fixed?

If $p(x) \in F[x]$ and $a,b$ are both roots of an irreducible polynomial $p(x),$ then $F(a) \cong F(b)$ by an isomorphism which takes $a$ onto $b$ and leaves every element of $F$ fixed. Simple ...