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)

-1
votes
2answers
601 views

What is $\operatorname{Gal}(\mathbb Q(\zeta_n)/\mathbb Q(\sin(2\pi k/n))$?

Let $\zeta_n$ be the $n$-th primitive root of unity and $4 \mid n$. Consider the field extensions $\mathbb Q \subset \mathbb Q(\sin(2\pi k/n) \subset \mathbb Q(\zeta_n)$. What is the degree of the ...
8
votes
4answers
732 views

Why is the product of all units of a finite field equal to $-1$?

Suppose $F=\{0,a_1,\dots,a_{q-1}\}$ is a finite field with $q=p^n$ elements. I'm curious, why is the product of all elements of $F^\ast$ equal to $-1$? I know that $F^\ast$ is cyclic, say generated by ...
7
votes
1answer
2k views

Existence of irreducible polynomials over finite field

Let $F$ be a finite field. How do we prove that for each $n \in \mathbb{N}$ there is an irreducible polynomial of degree $n$? One can assume that $F = \mathbb{F}_{p^m}$ where $p$ is prime. If $n \ge ...
5
votes
5answers
5k views

Finding inverse of polynomial in a field

I'm having trouble with the procedure to find an inverse of a polynomial in a field. For example, take: In $\frac{\mathbb{Z}_3[x]}{m(x)}$, where $m(x) = x^3 + 2x +1$, find the inverse of $x^2 + ...
4
votes
2answers
478 views

Is $\mathbf{Q}(\sqrt{2},\sqrt{3}) = \mathbf{Q}(\sqrt{6})$?

Is $\mathbf{Q}(\sqrt{2},\sqrt{3}) = \mathbf{Q}(\sqrt{6})$? If $\mathbf{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}\mid a,b,c,d\in\mathbf{Q}\}$ and $\mathbf{Q}(\sqrt{6})= \{a+b\sqrt{6} | ...
3
votes
2answers
91 views

Field that contains $(\mathbb{Z}/p^n\mathbb{Z})^*$

Let $p$ be an odd prime. There exists a field $F$ that contains an isomorphic copy of $(\mathbb{Z}/p^n\mathbb{Z})^*$ as a multiplicative subgroup? Clearly if $n=1$ we can take $F=\mathbb{F}_p$, but ...
3
votes
2answers
119 views

Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?

As title, Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated? Say if $K/F$ is a finite extension when $K$ is a finite-dimensional vector space of $F$. Clearly, ...
2
votes
2answers
560 views

Necessary and Sufficient Condition for a sub-field

Is there any necessary and sufficient condition to determine whether a subset $H$ of a given field $K$ is a subfield? In some paper I have found something like that: $H$ is a field if for all $a, ...
1
vote
1answer
110 views

Show that $9+9x+3x^3+6x^4+3x^5+x^6$ is irreducible given one of its roots

Given a polynomial $f(x)=9+9x+3x^3+6x^4+3x^5+x^6$ and one of its roots $\alpha=2^{1/3}+e^{2\pi i/3}$. Show that $f(x)$ is irreducible in $\mathbb Q[x]$. Eisenstein's criterion fails, it also didn't ...
1
vote
1answer
154 views

Minimal polynomial of intermediate extensions under Galois extensions.

Let $K$ be a Galois extension of $F$, and let $a\in K$. Let $n=[K:F]$, $r=[F(a):F]$, and $H=\mathrm{Gal}(K/F(a))$. Let $z_1, z_2,\ldots,z_r$ be left coset representatives of $H$ in $G$. Show that ...
5
votes
1answer
461 views

field of algebraic numbers

Would anyone be able to give me an outline or a hint towards the proof that the field of algebraic numbers is an infinite extension of the field of rationals? Many Thanks
5
votes
2answers
161 views

Closure of a number field with respect to roots of a cubic

Consider the following property of subfields ${\mathbb K}$ of ${\mathbb C}$ : $$ \text{Any polynomial of degree 3 with coefficients in} \ {\mathbb K} \ \text{has a root in } {\mathbb K} \ \ \ \ ...
5
votes
1answer
1k views

Sums and products of algebraic numbers

How does one go about proving that the sums and products of two algebraic numbers over a field $F$ (say $a,b\in K$, where $K/F$ is a field extension) is also algebraic? If we call $f_a$ and $f_b$ ...
2
votes
2answers
94 views

The set of algebraic numbers is a field.

I need help proving this stunning result. Let me state it this way. Let $\mathbb L$ be a field and $\mathbb K$ be a field with $\mathbb K \subset \mathbb L$ Let $A$ be the set of algebraic ...
2
votes
3answers
144 views

Explain why $\mathbb{Z \times Z}$ and $\mathbb{R \times R}$ is not a field [duplicate]

Explain why $\mathbb{Z \times Z}$ and $\mathbb{R \times R}$ is not a field and that why any external direct sum of two fields cannot be a field. I believe it has much to do with the lack of every ...
0
votes
3answers
290 views

is it possible for two fields with the same characteristic to not be isomorphic?

I know there is a long list of invariants, but is there a proof or theorem that explains two fields with the same characteristic are isomorphic?
26
votes
6answers
1k views

Is $\{0\}$ a field?

Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms: Closure under addition. ...
14
votes
4answers
4k views

Finding the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$.

I have to find the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$. The suggested way of doing it is to prove that $\mathbb Q[\sqrt 2 + \sqrt[3] 2]=\mathbb Q[\sqrt 2,\sqrt[3] 2]$ first. ...
26
votes
3answers
2k views

Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”?

The function $f(x)=x+\sin(x)$ is easily checked to be a bijection from the reals to itself, and so it has a unique inverse $y\mapsto g(y)$ such that $f\circ g=g\circ f$ are both the identity map. ...
11
votes
1answer
278 views

Field reductions

If there is a field $F$ that is a field reduction of the real numbers, that is $F(a)=\mathbb{R}$ for some $a$, let's also denote this $F=\mathbb{R}(\setminus a)$, then given $x \in \mathbb{R}$ is ...
10
votes
2answers
719 views

What exactly is the fixed field of the map $t\mapsto t+1$ in $k(t)$?

Suppose $k$ is a field, and $k(t)$ is the rational function field. If $f(t)=P(t)/Q(t)$ for some polynomials $P(t)$ and $Q(t)\neq 0$, then the map $t\mapsto t+1$ sends $f(t)$ to $f(t+1)$. So the ...
8
votes
5answers
431 views

Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$

I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
17
votes
5answers
504 views

Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we ...
7
votes
5answers
2k views

Infinite Degree Algebraic Field Extensions

In I. Martin Isaacs Algebra: A Graduate Course, Isaacs uses the field of algebraic numbers $$\mathbb{A}=\{\alpha \in \mathbb{C} \; | \; \alpha \; \text{algebraic over} \; \mathbb{Q}\}$$ as an example ...
6
votes
1answer
382 views

Degree of field extension

If $p$ is odd prime and $c=\cos(\frac{2\pi}{p})$, $s=\sin(\frac{2\pi}{p})$ then for which values of $p$ does $\mathbb{Q}(s,c)=\mathbb{Q}(c)$?
3
votes
2answers
616 views

isomorphism of Dedekind complete ordered fields

In the last chapter of Spivak's Calculus, there is a proof that complete ordered fields are unique up to isomorphism. I find the first steps in it somewhat suspicious. Specifically, I believe he is ...
10
votes
3answers
243 views

Does $\mathbb{F}_p((X))$ has only finitely many extension of a given degree?

We know that $\mathbb{Q}_p$ has only finitely many extensions of a given degree in its algebraic closure. Is it the same for $\mathbb{F}_p((X))$?
8
votes
2answers
973 views

Problem in Jacobson's Basic Algebra (Vol. I)

It is section $4.4$, exercise number $5$. It says the following: Let $F$ be a field of characteristic $p$. Show that $f(x)=x^p-x-a$ has no multiple roots and that $f(x)$ is irreducible in $F[x]$ if ...
6
votes
2answers
513 views

A field that is an ordered field in two distinct ways

Question: Explain the construction below (taken directly from Counter Examples in Analysis): An ordered field is a field $F$ that contains a subset $P$ such that $P$ is closed with respect ...
6
votes
1answer
396 views

Galois Groups of Finite Extensions of Fixed Fields

I am trying to prove the following proposition: Let $L$ be an algebraically closed field, $g \in Aut(L)$ and $K=\{x \in L \; | \; g(x)=x\}$. Show that every finite extensions $E/K$ is a cyclic ...
6
votes
2answers
434 views

Why aren't there any coproducts in the category of $\bf{Fields}$?

Well the question is stated in the title. I dont know much about field theory and i was suprised when i read it on wikipedia please provide some examples thanks in advance
5
votes
1answer
79 views

Why is the extension $k(x,\sqrt{1-x^2})/k$ purely transcendental?

Consider the function field $k(x,\sqrt{1-x^2})$ of the circle over an algebraically closed field $k$. Is $k(x,\sqrt{1-x^2})$ a purely transcendental extension over $k$? I'm curious because I was ...
4
votes
2answers
292 views

What is the condition for a field to make the degree of its algebraic closure over it infinite?

As we all know, the algebraic closure often has an infinite degree. Also, this shows the necessary and sufficient condition for a Galois extension to be a finite extension of fields. However, we may ...
3
votes
1answer
133 views

When is a divisible group a power of the multiplicative group of an algebraically closed field?

It is known that for any algebraically closed field $\mathbb{F}$ its multiplicative group $\mathbb{F}^*$ is a divisible group, and consequently any power $\mathbb{F}^*\times\cdots\times \mathbb{F}^*.$ ...
3
votes
2answers
425 views

A question about a weak form of Hilbert's Nullstellensatz

Corollary 5.24 on page 67 in Atiyah-Macdonald reads as follows: Let $k$ be a field and $B$ a finitely generated $k$-algebra. If $B$ is a field then it is a finite algebraic extension of $k$. We know ...
2
votes
2answers
792 views

Prove that every nonzero prime ideal is maximal in $\mathbb{Z}[\sqrt{d}]$

$d \in \mathbb{Z}$ is a square-free integer ($d \ne 1$, and $d$ has no factors of the form $c^2$ except $c = \pm 1$), and let $R=\mathbb{Z}[\sqrt{d}]= \{ a+b\sqrt{d} \mid a,b \in \mathbb{Z} \}$. ...
2
votes
1answer
324 views

a property of the radical closure of a field

Definition 1: Let $K$ be a field. If $\alpha$ is an algebraic element over K, such that $\alpha^n \in K $ and such that $x^n-\alpha^n \in K[x] $is also irreducible over K. Then we call $ K(\alpha)$ a ...
2
votes
1answer
538 views

If a normal $K/F$ has no intermediate extensions, then $[K : F]$ is prime

Let $K$ be a finite normal extension of $F$ such that there are no proper intermediate extensions of $K/F$. Show that $[K:F]$ is prime. Give a conterexample if $K$ is not normal over $F$.
0
votes
3answers
266 views

Proving that $-a=(-1)\cdot a$

As the title reveals, I want to prove (based on the axioms of field) that $$-a=(-1)\cdot a$$ I've been trying for a while now, but couldn't think of a way to do it and it got me thinking that maybe ...
7
votes
1answer
301 views

Generating Elements of Galois Group

I am trying to prove the following: Let $K=\mathbb{Q}(\sqrt{p_1},\ldots,\sqrt{p_n})$. Show that $K/\mathbb{Q}$ is Galois with Galois group $(\mathbb{Z}/2\mathbb{Z})^n$. I have attached my proof ...
7
votes
1answer
218 views

Field Extensions

Let $L/K$ a finite extension and $f(x)\in K[x]$ a non-linear irreducible polynomial. Prove that if $\mathrm{gcd}\left( \mathrm{deg}(f) , \left[ L:K \right] \right)=1$ then $f(x)$ has no roots in ...
6
votes
1answer
374 views

Does $\varphi(1)=1$ if $\varphi$ is a field homomorphism?

Is it by definition that $\varphi(1)=1$ if $\varphi$ is a field homomorphism ? My field theory lecture said that yes, but now $\varphi\equiv0$ is not a field homomorphism... What is the 'standard' ...
5
votes
1answer
242 views

Finite Field Extensions and the Sum of the Elements in Proper Subextensions (Follow-Up Question)

I recently posted the following question, to which this question is a follow-up. Regardless, my question here will be self-contained. Let $F$ be a finite field, and let $u,v$ be algebraic over $F$. ...
4
votes
1answer
1k views

What's the difference between hyperreal and surreal numbers?

The Wikipedia article on surreal numbers states that hyperreal numbers are a subfield of the surreals. If I understand correctly, both fields contain: real numbers a hierarchy of infinitesimal ...
4
votes
2answers
473 views

The endomorphism of field

Can we find a field $K$ and an endomorphism $f$ of $K$, such that $f$ is not trivial and $f$ is not surjective? In other words, can we find an endomorphism of $K$ which is not an automorphism?
4
votes
1answer
275 views

A proof of the Lagrange's theorem on cyclic extention fields

I came up with the following proof of the above theorem which is the key to the Galois's theory of algebraic equations. The usual proof uses Lagrange resolvent or Hilbert 90 which uses a similar ...
4
votes
2answers
726 views

Exercise on separable polynomials over fields of prime characteristic

Having learned about separable polynomials today in class, I tried to do the following exercise concerning separable polynomials, namely: Suppose $f$ is the minimal polynomial of $a$ over a field ...
4
votes
3answers
1k views

Example of a complete, non-archimedean ordered field

I'm looking for a concrete example of a complete (in the sense that all Cauchy sequences converge) but non-archimedean ordered field, to see that these two properties are independent (an example of ...
3
votes
1answer
106 views

Fixed Field of Automorphisms of $k(x)$

Fixed field of automorphisms of $k(x)$, with $k$ a field, induced by $I(x)=x$, $\varphi_1(x) = \frac{1}{1-x}$, $\varphi_2 (x)=\frac{x-1}{x}$? Since $I(x)=x$, $\varphi_1(x)=\frac{1}{1-x}$, ...
3
votes
1answer
266 views

A formula for the roots of a solvable polynomial

Let $F$ be a field and $p(x)\in F[x]$ a separable polynomial, denote $K$ as the splitting field of $p$ and assume that $K/F$ is Galois with a solvable Galois group. I don't understand if this imply ...