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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4
votes
3answers
494 views

Proving that $x^{2}+2$, $x^{2}-x+4$, and $x^{3}+3x-1$ are irreducible over $\mathbb{Q}$

Let $f$, $g$ and $h$ be the polynomials given by: $$f(x)=x^{2}+2$$ $$g(x)=x^{2}-x+4$$ $$h(x)=x^{3}+3x-1$$ Show that $f$, $g$ and $h$ are irreducible over $\mathbb{Q}$. I do this: ...
1
vote
1answer
205 views

splitting field of $x^p-a$ over $\mathbb{Q}$ has no primitive $p^2$ roots of unity

It is known that the splitting field of $x^p-a$ over $\mathbb{Q}$ has no $p^2$ roots of unity. We can assume $a\in \mathbb{Q}$ is not a pth power in $\mathbb{Q}$. I came up with the following proof of ...
7
votes
2answers
386 views

Compositum of fields with trivial intersection

Let $E/F$ be a finite extension. Let $L,K$ be two intermediate fields with $L\cap K = F$, and also $$[L : F] [K:F] = [E:F].$$ Must it hold that the compositum $LK$ equals $E$? If we assume that $E/F$ ...
5
votes
4answers
599 views

$F:= \{ a+b\sqrt{7} \mid a,b \in \mathbb{Q} \}$ closed under addition, subtraction, multiplication, and division

I am in my math class and I came across this problem on my past midterm. How can we prove that $F:=\{ a+b\sqrt{7} \mid a,b \in \mathbb{Q} \} $ is closed under addition, subtraction, multiplication, ...
4
votes
1answer
65 views

Polynomial Fields.

I am trying to find a polynomial in $\Bbb{Q}[x]$ which is irreducible over $\Bbb{Q}$ and has at least one linear factor over $\Bbb{R}$ and at least one irreducible quadratic factor over $\Bbb{R}$. Any ...
1
vote
2answers
259 views

Monomorphisms of a finite field extension

I have to show that if $L:K$ is a finite field extension and we have a $K$-monomorphism then this is an automorphism. I'm a bit confused by this if we have a k monomorphism: $f_K:L\rightarrow G$ for ...
5
votes
1answer
138 views

Field Extensions Problem - From Paolo Aluffi's Book

This is the exercise 4.7, chapter VII, from Paolo Aluffi's algebra book. I'm sorry for just copying the question without writing any development myself, I don't have a single ideia about how to use ...
9
votes
1answer
203 views

How to compute the Galois group of $x^5+99x-1$ over $\mathbb{Q}$?

I was stuck trying to compute the Galois group of $x^5 + 99x -1$. The problem asks to compute the Galois group over $\mathbb{F}_2, \mathbb{F}_3, \mathbb{F}_5, \mathbb{F}_{11}$ and $\mathbb{Q}$. I was ...
4
votes
4answers
126 views

Why do $f$ and $f'$ generate all of $K[X]$?

I have been studying Marcus' Number Fields. I am stuck on a remark in Appendix 2, page 258. He says: A monic irreducible polynomial $f$ of degree n over $K$ (a subfield of $\mathbb{C}$) splits into n ...
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votes
3answers
71 views

A sum of products symmetric in the images under all the embeddings

Let $\mathbb{Q}\subset K\subset \mathbb{C}$ where $K$ is a finite extension of $\mathbb{Q}$. Let $\sigma_1, \dots, \sigma_n$ be all the embeddings $K\rightarrow \mathbb{C}$. Is it true that elements ...
2
votes
1answer
55 views

Algebraic numbers and fraction field.

If $a\in\mathbb{C}$ is algebraic, then $\mathbb{Q}(a)=\mathbb{Q}[a]$, and the converse holds too. I am having trouble proving that the same holds for several elements. Is it true that $a_1,\ldots, ...
2
votes
2answers
308 views

order of elliptic curve $y^2 = x^3 - x$ defined over $F_p$, where $p \equiv 3 \mod{4}$

It is said that the elliptic curve $y^2 = x^3 - x$ defined over a prime field $\mathbb{F}_p$, where $p \equiv 3 \mod{4}$ has an order $p + 1$. When I tried to get the elements of $E = \{(x,y) \in ...
4
votes
1answer
835 views

How to find all automorphisms of $\mathbb{Q}(\sqrt[3]{5})$? [duplicate]

Find all automorphisms of $\mathbb{Q}(\sqrt[3]{5})$. How can I solve the above problem ? Please help someone.
4
votes
1answer
252 views

Multiple roots of polynomials over a finite field

Show that $x^4+x+1$ over $\mathbb{Z}_2$ does not have any multiple zeros in any field extension of $\mathbb{Z}_2$. Show that $x^{21} + 2x^8 +1$ does not have multiple zeros in any ...
6
votes
6answers
2k views

Abstract algebra book recommendations for beginners.

I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved ...
2
votes
1answer
596 views

Finding a primitive element for a field extension

Find a primitive element for the field extension $\mathbb Q(2^{1/3},2^{1/4})$ of $\mathbb Q$
4
votes
3answers
635 views

Irreducible $f(x) \in F[x]$ of prime degree, $E/F$ finite extension, $p \mid [E:F]$.

Let $F$ be a field and let $f(x) \in F[x]$ be irreducible of prime degree $p$. Let $E/F$ be a finite extension. Prove: If $f(x)$ is not irreducible in $E[x]$, then $p \mid [E:F]$. (Hint: Consider a ...
2
votes
1answer
118 views

Multiplication in $\mathbb{R}^n$

I have read that it is not possible to define multiplication in $\mathbb{R}^n$ for $n\ge 3$ in any manner whatsoever so that together with usual addition it forms a field. However I have not been able ...
3
votes
4answers
87 views

Prove that $f\sim g$ iff $f\mid g$ and $g\mid f$, where $f$ and $g$ are polynomials.

Prove that $f\sim g$ iff $f\mid g$ and $g\mid f$, where $f$ and $g$ are polynomials. It seems to me that $f$ and $g$ must be equal. And if you guys could direct me to sources that explain ...
1
vote
3answers
518 views

Two finite fields are isomorphic. [duplicate]

Let $F = \Bbb{Z}_2$. Given the irreducible polynomials $f(x)= x^3 + x + 1$, and $g(y) = y^3 + y^2 + 1$, form the fields $K = F[x]/(f(x))$ and $E = F[y] / (g(y))$. These are fields of order 8 ...
5
votes
1answer
252 views

Euclidean norms in quadratic fields

I'm currently reading a set of lecture notes of Number Theory, and there's a small part I'm having trouble understanding. A norm $N: R \rightarrow \mathbb{N} $ is Euclidean if it satisfies: for ...
0
votes
1answer
51 views

Elements in extension field

Suppose $\alpha\in\text{GF}(q^n)\setminus\text{GF}(q)$. Then there exists an irreducible polynomial $f(x)\in\text{GF}(q)[x]$ such that $f(\alpha)=0$. My question is that whether this $f(x)$ is unique ...
9
votes
3answers
2k views

How to show that a finite commutative ring without zero divisors is a field?

$R$ is finite commutative ring without zero divisors which has at least two elements. How to show that $R$ is field? I'm just starting with abstract algebra and I'd really appreciate if someone ...
3
votes
2answers
182 views

Final object in fields of characteristic $ 0 $?

In his answer to this question: Category of Field has no initial object, Arturo Madigin indicated that the field of rational numbers is the initial object in the category of fields of characteristic $ ...
4
votes
2answers
528 views

Can nonzero polynomials vanish identically?

I know that a nonzero single-variable polynomial over a finite field can vanish identically e.g. take the product $\prod_a(x-a)$ for every $a$ in the field. But I know that for an infinite field this ...
2
votes
1answer
214 views

Number of monomorphisms $\mathbb{Q} \to \mathbb{C}$

In my abstract algebra class, we have been tasked to find all monomorphisms $\mathbb{Q} \to \mathbb{C}$. The book (Stewart's Galois Theory) gives an example for $\mathbb{Q}(\sqrt[3]{2}) \to ...
3
votes
1answer
278 views

Calculate $\mathrm{Gal}(\mathbb{Q}(\sqrt[5]{3})/\mathbb{Q})$

I'm attempting some of my first problems in solving for Galois Groups, and this one has stumped me. What I've done so far is found that $\mathbb{Q}(\sqrt[5]{3})$ is not a normal extension, because ...
4
votes
1answer
132 views

Help with a bilinear form

Let $a,b\in\mathbb{F}_{2^{m}}$ (a field of characteristic $2$, m is odd) I need to prove that ...
5
votes
2answers
124 views

Trace function equation

Let $a,b\in\mathbb{F}_{2^{m}}$ (a field of characteristic $2$, m odd), with $a,b\neq 0$. I need to prove that $$\sum_{i=1}^{(m-1)/2}\operatorname{tr}(a^{2^{i}}b+b^{2^{i}}a)=0\qquad \text{ iff }\qquad ...
5
votes
1answer
450 views

Calculating Splitting Field Degree of Extension

Is there an easier way to calculate the degree of extension of a splitting field for a polynomial like $$x^7-3\quad\text{over}\quad\Bbb Z_5\,?$$My approach for several of these have been to find all ...
3
votes
2answers
94 views

Why does each subfield have to contain multiples of 1?

I have seen a Theorem: every field contains exactly one prime subfield $K_0$. and next: $K_0$ has to contain 1 and all its multiples: $n \cdot 1 = 1 + \ldots + 1$ Is this because it has to ...
2
votes
2answers
61 views

Is $N_{k\subset K}$ the only *norm* on the field extension $k\subset K$?

In several examples of field extensions the norm function is very useful. For instance, in $\mathbb{Q}\subset\mathbb{Q}(\sqrt{2})$, the norm is $N(x+y\sqrt{2})=x^2-2y^2$. In ...
7
votes
4answers
762 views

Show $x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q[x]$.

Show $p(x) = x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q[x]$. By Gauss' Lemma, a primitive polynomial in $\mathbb Z[x]$ is irreducible in $\mathbb Q[x]$ if and only if it is irreducible in ...
3
votes
1answer
198 views

Subfield of rational function fields

Let $k\subset F\subseteq k(x_{1},x_{2},...,x_{n})$ where $k$ and $F$ are the fields and $x_{1},x_{2},...,x_{n}$ are transcendental over $k$. Can we express $F$ in terms of or function of ...
1
vote
4answers
144 views

Let $\mathbb{F}$ be any field. Show that the number of cube-roots of unity in $\mathbb{F}$ is either $1$ or $3$.

Let $\mathbb{F}$ be any field. Show that the number of cube-roots of unity in $\Bbb F$ is either $1$ or $3$. Show that if $\mathbb{F}$ has characteristic $3$ then it has only one cube-root of ...
1
vote
2answers
63 views

Struggling with a question on quotients in elementary Galois theory

I have started teaching myself Galois Theory. I have a problem understanding a part of the proof of the following proposition : Let $K\subseteq L$ be a field extension and $l\in L$ an element which is ...
1
vote
2answers
158 views

Subfields of Rings

I am currently working through an undergraduate class in Galois Theory. I have come across a question that I am unsure about. Can a ring that is not a field, have a subring that satisfies the ...
5
votes
2answers
155 views

Transcendental extension of field.

Let $F(a)$ be a transcendental extension of the field $F$. Given an element $b \in F(a)$ such that $b \notin F$, I would like to show that $F(a)$ is an algebraic extension of $F(b)$. My idea of ...
3
votes
2answers
227 views

If $\mathrm{char}(K)=p$ is prime, $L/K$ is separable if and only if $K(\alpha) =K(\alpha^p)$ for all $\alpha \in L$ [duplicate]

I am trying to prove that if $L/K$ is an algebraic extension and if $\alpha \in L$, then $\alpha$ is separable over $K$ if $\mathrm{char}(K)=0$. This is clear because $K$ is perfect which in turn ...
3
votes
1answer
86 views

When are powers of primitive elements still primitive elements

This question is motivated by this question and is tangentially related to this question. Let $L/K$ be a finite Galois extension of fields. Pick $\alpha \in L \setminus K$ and consider the simple ...
2
votes
1answer
117 views

Problem with roots of unity

Let $\zeta$ a root of $x^{p}-1$, with $p$ an odd prime, and $K$ a subgroup of the mutiplicative group $\mathbb{Z}_p^{*}$ of index $2$. I need to prove that $a=\displaystyle\sum_{k\in K}\zeta^{k}$ ...
1
vote
2answers
243 views

Transcendental elements over field extensions.

Let $E/F$ be a field extension, and suppose $a \in E$ is transcendental over $F$. I'm reading a proof that says $\dim_F F(a) \ge \dim_FF[x] = + \infty$ since the evaluation map $F[x] \to F [a]$, ...
2
votes
1answer
129 views

Injection from an integral domain to its field of fractions.

I have a quick question about modules. Suppose that $R$ is an integral domain with field of fractions $K$. Then any free $R$-module is isomorphic to copies of direct sums of $R$, say $R^i$ . ...
2
votes
1answer
252 views

Countable Field Extension of a Countable Field

Okay, first question on this site, apologies in advance for any mistakes I may make. Question: So I need to show that an algebraic field extension $E:F$, with $F$ being countable, is countable. My ...
3
votes
3answers
572 views

Field extensions that are not normal

I am trying to come up with field extensions $M : L : K$ such that none of the three extensions $M:L, L:K, M:K$ are normal. So far, I have tried letting $K = \mathbb{Q}, L = \mathbb{Q}(\sqrt[3]{2})$. ...
7
votes
2answers
218 views

Explicit Galois theory computation in cyclotomic field

Let $p$ be an odd prime. Let $\zeta=e^{\frac{\pi i}{4 p}}$, thus $\zeta$ is a primitive $8p$-th root of unity. There is a unique $\mathbb Q$-automorphism $\tau$ of the number field ${\mathbb ...
2
votes
4answers
465 views

Polynomial factorization over finite fields

How can i factorize the polynomial $x^{12}-1$ as product of irreducibles polynomials over $\mathbb{F}_4$? Anyone can help me?
6
votes
1answer
456 views

Irreducible cyclotomic polynomial

I want to know if there is a way to decide if a cyclotomic polynomial is irreducible over a field $\mathbb{F}_q$?
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vote
2answers
224 views

Galois group of a polynomial

I want to know how to find a polynomial $f(x)$ of degree $5$ in $\mathbb{Q}[x]$ with Galois groups $G_f=\mathbb{S}_5$
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votes
6answers
1k views

Is $\mathbf{C}$ the algebraic closure of any field other than $\mathbf{R}$?

It seems to me (intuitively) that there should be no other fields whose algebraic closure is $\mathbf{C}$, even though I have no reason to believe it. The facts I've been using to formulate an ...