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 ...

learn more… | top users | synonyms (1)

2
votes
1answer
32 views

Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$?

Consider the ring $M_2(\mathbb{R})$ of all 2 × 2 real matrices over $\mathbb{R}$. Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$? Is the quotient ring ...
-3
votes
0answers
30 views

Whether the number $a$ is algebraic over $\mathbb{Q}$ [on hold]

Is the number $a = \displaystyle \left( \frac{(1 + \sqrt[3]{7})^{\tfrac{7}{5}}}{(\sqrt[3]{7} - 7)^3 + 77} \right)^{13}$ algebraic? If so, is algebraic degree of $a$ bounded by $15$?
0
votes
0answers
9 views

Problem of composite and finite fields.

Problem: Let $L$ and $M$ be intermediate fiels of the extension $K \subset F$, of finite dimension over $K$. Assume $[LM:K]=[L:K][M:K]$ and prove that $L \cap M =K$. Must try $[L \cap M:K]=1$. We ...
0
votes
1answer
10 views

Polynomial with degree less than degree of an irreducible polynomial of the same root is 0

Let $F$ be a field, and $p(x)\in F[x]$ be an irreducible polynomial. Suppose $\alpha$ is a root of $p(x)$. Show that if $q(x)$ is a polynomial such that $\deg q(x) < \deg p(x)$ and $q(\alpha)=0$, ...
0
votes
2answers
21 views

Relation between reduced finite algebra, prime ideal and field extension

Is it true that if $L$ is a reduced finite dimensional commutative algebra over a field $K$ (which is finite or of characteristic $0$) and if $\mathfrak p$ is a prime ideal of $L$, then ...
0
votes
4answers
60 views

Characteristic of a Finite Integral Domain

I am a little confused as how to approach this problem. The title of this problem is the title of the section which it comes from. However, there is no information that the given integral domain is ...
0
votes
1answer
16 views

I need to show that if $K$ is of characteristic $0$,the algebra $A$ has a primitive generator.

Let $K$ be a field and $A$ a reduced K-algebra of finite dimension over $K$. I need to show that if $K$ is of characteristic $0$, $A$ has a primitive generator (i.e. $A=K[x], x \in A$) I've proved ...
0
votes
2answers
24 views

Why is the fixed field of this automorphism is $Q(\pi^2)$?

Let $\sigma:Q(\pi)\rightarrow Q(\pi)$ be an auorphism fixing $Q$ such that $\sigma(\pi)=-\pi$. Let $F$ be the fixed field. Then it is obvious that $Q(\pi^2)\subset F$. However, why Why is ...
2
votes
2answers
64 views

How do I show that $\sqrt{5}\notin \mathbb{Q}(\sqrt{2},\sqrt{3})$?

How do I show that $\sqrt{5}\notin \mathbb{Q}(\sqrt{2},\sqrt{3})$? Since $X^2-5$ is the minimal polynomial of $\sqrt{5}$ over $\mathbb{Q}$ and its degree is not relatively prime to ...
0
votes
2answers
29 views

$x^{p^{k}-1}-1$ divides $x^{p^{n}-1}-1$ in $\mathbb{F}_{p}$ iff $k$ divides $n$

Let's consider two polynomials in $\mathbb{F}_{p}[x]$: $f(x)=x^{p^{n}-1}-1$ and $g(x)=x^{p^{k}-1}-1$. How to prove that $g(x) \mid f(x)$ iff $k \mid n$? For instance, it's worth trying this: ...
0
votes
0answers
24 views

Proof of primitive element theorem for $F$ finite

I am trying to understand the proof of primitive element theorem, in particular this statement: Let $E/F$ be a field extension such that there are finitely intermediate fields containing $F$. Then $E ...
2
votes
1answer
22 views

Algebraic subfield of transcendental extension

I was recently thinking about whether it is possible to generate an infinite dimensional algebraic extension over a base field using just finitely many transcendental elements. Specifically, given a ...
0
votes
1answer
24 views

How do I prove that $X^{p^n}-X$ is the product of all monic prime polynomials of degree dividing $n$?

How do I prove that $X^{p^n}-X$ is the product of all monic prime polynomials in $Z_p[X]$ of degree dividing $n$? Let $\bar Z_p$ be an algebraic closure of $Z_p$. Define $F=\{x\in \bar ...
0
votes
0answers
65 views

Is this element algebraic over $\mathbb{Q}$?

I'm trying to see whether the following element is algebraic over $\mathbb{Q}$ and if it is - try and find its degree: $$ a = \left(\frac{(1 + \sqrt[3]{7})^{7/5}}{(\sqrt[3]{7} - 7)^3 + ...
1
vote
0answers
16 views

Help in the proof of $k \subset F$ is radical if and only if it is solvable.

Let $k \subset F$ be a Galois Extension, with char $k=0$. Provided it has enough roots of $1$, $k \subset F$ is radical if and only if it is solvable. I am trying to show that Solvability $\implies$ ...
0
votes
1answer
23 views

Construct the Normal Closure for this extension

I am trying to find the normal closure for the following extension $\Bbb Q\subset\Bbb Q(t)$, where "$t$" is a zero of $x^3-3x^2+3$ and $\Bbb Q$ are the rational numbers. I know the normal closure ...
2
votes
1answer
31 views

Proving that the unit cube cannot be tripled (with straight edge and compass)

I would like to show the unit cube cannot be tripled using a straightedge and compass. I note that the side of a cube that has been tripled would have a side length of $\alpha=\sqrt[3]{3}$ But ...
1
vote
0answers
28 views

Identifying a group

When asked to identify the group $\mathbb{F}_4^{+}$, is my explanation below complete? If not, how can I complete it? The field $\mathbb{F}_4^{+} = \mathbb{F}_2[x]/(x^2+x+1)$ consists of the residues ...
0
votes
3answers
29 views

Finite fields and isomorphism

For each prime number p, let $F_p$ denote the field of integers modulo p. Now let K be any finite field. a) Prove that K contains a subfield isomorphic to $F_p$ for some prime number p b) Prove that ...
1
vote
0answers
20 views

On the existence of a polynomial in several variables with only one zero

Given an ordered field $\mathbb{K}$, then for all $n>1$ there exists $f\in\mathbb{K}[X_1,\ldots,X_n]\ s.t.\ \mathcal{V}^{\mathbb{K}^n}(f):=\{p\in\mathbb{K}^n:f(p)=0\}=\{(0,\ldots,0)\}$ For ...
0
votes
1answer
36 views

Squares in a field with $q^n$ elements.

Let $F$ be a field with $q^n$ elements, where $q$ is an odd prime. Write $q^n=2m +1$ with $m \in \mathbb{N}.$ If $r \in F^{\times},$ show that the equation $y^2= r$ has a solution iff $r^m=1.$
3
votes
2answers
27 views

How do I prove that $\forall \beta\in F(\alpha)\setminus F$ is transcendental?

Let $E/F$ be a field extension. Let $\alpha\in E$ be transcendental over $F$. Let $\beta\in F(\alpha)\setminus F$. Then, how do I prove that $\beta$ is transcendental over $F$? Here's how I tried: ...
3
votes
0answers
40 views

“Breaking the symmetry” in solving algebraic equations

I've heard somewhere a discussion about solving algebraic equations before: When solving a quadratic equation, we are essentially doing the following. Observe that ...
2
votes
1answer
29 views

$a$ and $1+a^{-1}$ have same degree over $F$ if $a$ is algebraic over $F$

Suppose that $a$ is algebraic over a field $F$. Show that $a$ and $1+a^{-1}$ have the same degree over $F$. Not really sure where to start for this one. I know that I have to show that ...
0
votes
1answer
12 views

What is an example of $E/F,L/E$ are normal but $L/F$ is not. [duplicate]

Let $E/F,L/E$ be normal field extensions. What would be an example such that $L/F$ is not normal?
0
votes
2answers
35 views

Let $a$ be a complex zero of $x^2+x+1$ over $\mathbb{Q}$. Prove that $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(a)$.

Let $a$ be a complex zero of $x^2+x+1$ over $\mathbb{Q}$. Prove that $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(a)$. Let $f(x)=x^2+x+1$. Then $f(a)=a^2+a+1=0$. To show equality of the two fields, we need ...
1
vote
2answers
50 views

How to find the degree of an extension field?

How to find the degree of an extension field ? Let $f:=T^3-T^2+2T+8\in\mathbb Z[T]$ and $\alpha$ be the real root of $f$. Why is then $\mathbb Q(\alpha)$ is a number field of degree $3$ ? I've ...
1
vote
2answers
14 views

What is an example of transcendental extension such that a monomorphism cannot be extended?

Let $E/F$ be a field extension and $\alpha\in E$ be transcendental over $F$. Let $\bar F$ be the algebraic closure of $F$ and $\sigma:F\rightarrow \bar{F}$ be a field monomorphism. What is an ...
1
vote
2answers
28 views

How do I find all conjugates of $\sqrt{1+\sqrt{2}}$ over $\mathbb{Q}$?

How do I find all conjugates of $\sqrt{1+\sqrt{2}}$ over $\mathbb{Q}$? I have shown that $\sqrt{1+\sqrt{2}}$ is a root of $X^4 - 2X^2 - 1$ and this polynomial is irreduciable in $\mathbb{Q}[X]$, ...
2
votes
2answers
37 views

Infinite subspaces for a vector space that cannot be spanned by a single element

If a vector space (over an infinite field) cannot be spanned solely by a single element, does it mean it has infinite subspaces? I couldn't find an example that contradicts this
2
votes
0answers
45 views

Showing that $40^{\circ}$ is not constructible

Show that $40^{\circ}$ is not constructible. Attempt We note that $\cos 120^{\circ}=-\frac{1}{2}$ and that it also equals $4\cos^340^{\circ}-3\cos40^{\circ}$, which is obtained by using the ...
1
vote
0answers
19 views

What would be a value of $X$ under an automorphism of $F(X)$ over $F$?

Let $\sigma:F(X)\rightarrow F(X)$ be a field automorphism fixing $F$. What would be an value of $\sigma(X)$? Since $X$ is transcendental over $F$, $\sigma(X)$ is a transcendental over $F$ and ...
0
votes
1answer
18 views

Q basis for splitting field

I have the following field theory question: I am given this polynomial $ x^5-5 $ for which I am supposed to find a basis for the splitting field over Q all I can determine in this regard is that it ...
0
votes
1answer
18 views

prove $[E(a):E] \le [F(a):F]$

Let $F<E<K$ be field extensions, such that $a \in K$, and $[K:F]<\infty$, Is it true that $[E(a):E] \le [F(a):F]$? How can I show this?
1
vote
1answer
11 views

Are these two definitions of separable extension equivalent?

Definition1 - wikipedia Let $E/F$ be an algebraic extension. Then $E/F$ is separable iff for each $\alpha\in E$, the minimal polynomial of $\alpha$ over $F$ is separable. Definition 2 - ...
1
vote
2answers
64 views

Is there an infinite field such that every non-zero element has finite multiplicative order?

Is there an infinite field such that every non-zero element has finite multiplicative order? I did not find any example of such a field, but also did not see anything that forbids the existence of ...
2
votes
1answer
19 views

Looking for different proofs for $-1$ is a quadratic residue of primes of the form $4k+1$ and related facts

Suppose $p$ is a prime of the form $4k+1$ , then $4|p-1=|\mathbb Z_p^*|$ , as $\mathbb Z^*_p$ is a cyclic group , so there is $\bar x \in \mathbb Z_p^*$ such that $o(\bar x)=4$ , then $o(\bar x^2)=2$ ...
1
vote
1answer
21 views

Question about field extension notation

Hello all I was given the following question about which I understand everything except possibly the notation. I am given a sub-field $ F \subseteq R $ and I am asked to prove the degree of the "field ...
1
vote
1answer
20 views

Prove that there are no separable extensions of $k$ of degree $n$

Let $k$ be a field and let $n \gt 0$ be an integer. Assume that there are no irreducible polynomials of degree $n$ in $k[x]$ . Prove that there are no separable extensions of $k$ of degree $n$ I ...
13
votes
2answers
2k views

Proving the “freshman's dream”

$(x+y)^p = x^p + y^p$ holds in any field of characteristic $p$. However all the proofs I have seen use induction and some relatively nasty algebra despite how fundamental this fact seems. What is the ...
0
votes
2answers
20 views

Show that if $F$ is a field, then $<x>$ is maximal in $F[x]$. Also, show that $F[x]$ is not local.

See statement above. So far I have the following: Assume that $<x>$ is not maximal. Then $ <x> \subset <f(x)> \neq F[x]$. This means that $x = f(x) g(x)$. Since $x$ is ...
0
votes
1answer
11 views

About relation between degree of extension and normality

Hi i know that if $F<E$ is an field extension and $[E:F]=2 $ then it is normal extension. How can i show that if $[E:F]=k$ for any $k>2 $ doesnt imply E is normal extension. Basically i need ...
1
vote
2answers
62 views

Field $K (x)$ of rational functions with coefficients from $K$, if $f\in K(x)$, then $f^2 \neq x^2-1$

I'm in the process of studying for an exam and I came across the following question: Prove that if $K$ is a field and $K (x)$ is the field of rational functions with coefficients from $K$, if ...
3
votes
2answers
48 views

Prove that $x^2 + 3x +2$ is irreducible in $\mathbb{Z}[[x]]$, but not in $\mathbb{Z}[x]$.

As the problem states, I need to show irreducibility of the given polynomial. I'm not sure where go with this, so any help would be great. I know that Eisenstein has a nice test for this in ...
0
votes
1answer
27 views

How's this inertia called?

Let $E/F$ be an algebraic extension. Let $L_1,L_2$ be algebraically closed fields and $\sigma_1:F\rightarrow L_1,\sigma_2:F\rightarrow L_2$ be field monomorphisms. Define ...
0
votes
2answers
24 views

Square root constructible elements in a splitting field. [on hold]

Let α have minimal polynomial p(x) ∈ $\mathbb{Q}$[x], with roots α = α1, ..., αs. Let K = $\mathbb{Q}$(α1, ..., αs) be the splitting field of p(x). Let [K:Q] = 2w for some integer w. Prove ...
2
votes
2answers
24 views

Does every algebraically closed field with nonzero characteristic have a unique finite subfield $p^n$?

Let $F$ be an algebraically closed field such that $char(F)\neq 0$. Then, $\forall n\in\mathbb{Z}^+$, there exists a unique finite subfield $K$of $F$ such that $|K|=char(F)^n$. Is this ...
2
votes
0answers
35 views

Notation Question in proof of Kronecker-Weber theorem

I am trying to understand this proof of the Kronecker-Weber theorem by Franz Lemmermeyer http://arxiv.org/pdf/1108.5671.pdf. The set-up is this: $K/\mathbb{Q}$ is a cyclic extension of prime degree ...
0
votes
0answers
43 views

Questions about the function fields of complex algebraic surfaces

Let $X$, $Y$ be complex algebraic surfaces(Of course, they are smooth). Suppose that $X$ is normal. Let $K(X)$ and $K(Y)$ be the function fields of $X$ and $Y$, respectively. And we have a ...
1
vote
0answers
43 views

A problem about an algebraic number [duplicate]

Show that $2^{\frac{1}{2}}+5^{\frac{1}{3}}$ is algebraic over $\mathbb{Q}$ of degree $6$. Can I just construct $x=2^{\frac{1}{2}}+5^{\frac{1}{3}}$, $x-2^{\frac{1}{2}}=5^{\frac{1}{3}}$, ...