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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25
votes
4answers
1k views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
15
votes
4answers
3k 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. ...
11
votes
1answer
274 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 ...
6
votes
1answer
362 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 $\mathbb{Q}(s,c)=\mathbb{Q}(c)$?
6
votes
2answers
347 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
1
vote
1answer
390 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 $F$.
22
votes
3answers
472 views

On the meaning of being algebraically closed

The definition of algebraic number is that $\alpha$ is an algebraic number if there is a nonzero polynomial $p(x)$ in $\mathbb{Q}[x]$ such that $p(\alpha)=0$. By algebraic closure, every nonconstant ...
9
votes
3answers
157 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 $\mathbb{F}_p((X))$?
7
votes
5answers
1k 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 ...
5
votes
1answer
345 views

Irreducible cyclotomic polynomial

I want to know if there is a way to decide if a cyclotomic polynomial is irreducible over a field $\mathbb{F}_q$?
4
votes
2answers
275 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
120 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
359 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
1answer
263 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 ...
0
votes
3answers
257 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 ...
-1
votes
2answers
539 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 ...
13
votes
2answers
253 views

Sum of irrational numbers, a basic algebra problem

Let $x_1,\dots,x_n$ be positive rational numbers. If $\sqrt[l_1]{x_1},\dots,\sqrt[l_n]{x_n}$ are all irrational numbers (where $l_1,l_2,\dotsc,l_n\in\Bbb N^*$), does it follow that $$\sqrt[l_1]{x_1}+ ...
7
votes
1answer
1k 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 ...
7
votes
1answer
200 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
3answers
837 views

If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field. [duplicate]

Possible Duplicate: Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field I need to prove this result, but the only starting point I think of is to ...
4
votes
1answer
224 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
634 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 ...
3
votes
1answer
750 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 ...
3
votes
1answer
165 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 ...
3
votes
3answers
599 views

Primitive polynomials of finite fields

there are two primitive polynomials which I can use to construct $GF(2^3)=GF(8)$: $p_1(x) = x^3+x+1$ $p_2(x) = x^3+x^2+1$ $GF(8)$ created with $p_1(x)$: 0 1 $\alpha$ $\alpha^2$ $\alpha^3 = ...
1
vote
2answers
425 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 $R=\mathbb{Z}[\sqrt{d}]= \{ a+b\sqrt{d} \mid a,b \in \mathbb{Z} \}$. ...
1
vote
1answer
47 views

Reducibility over a certain field.

Let $K=F_2[x]/(x^3+x+1)$. I want to show that $f(x)=x^4+x^2+1$ is reducible over $K$ but has no roots in it. How to proceed? I know that $F$ contains 8 elements, how is the structure of these ...
1
vote
2answers
365 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 ...
6
votes
1answer
251 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 ...
6
votes
2answers
272 views

Field Isomorphisms

Suppose $F/L$, $F'/L$, $L/K$ finite extensions of fields. If $F$, $F'$ isomorphic over $K$ then does it follow that they are isomorphic over $L$? I think probably not, but I can't come up with a ...
6
votes
3answers
221 views

What's the difference between $\mathbb{Q}[\sqrt{-d}]$ and $\mathbb{Q}(\sqrt{-d})$?

Sorry to ask this, I know it's not really a maths question but a definition question, but Googling didn't help. When asked to show that elements in each are irreducible, is it the same?
6
votes
1answer
328 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
2answers
164 views

Towers of perfect fields of mostly order $p^t$

This question is strongly related to this question. I'm curious about the following result. Fix a prime $p$ and let $L$ be a perfect field and $q_1(t),\dots,q_n(t) \in L[t]$ irreducible polynomials ...
4
votes
1answer
233 views

Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$

Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$. My justification for this question is as follows; Suppose $F(\alpha^2)\subsetneq F(\alpha)$, we have $F \subsetneq F(\alpha^2) ...
4
votes
3answers
384 views

Equivalence of Archimedian Fields Properties

I'm trying to prove that an Archimedian field is a subfield of the real numbers, my plan is to use the fact that the rationals are dense within the field and their Dedekind completion is the real ...
3
votes
0answers
123 views

Link between representation theory and Galois theory: Trivial representation in field towers.

Let $K|F$ be a finite cyclic Galois extension of number fields of degree prime to $p$ with Galois group $H$, where $p$ denotes a rational prime. Let $L|K$ denote a pro-$p$-extension (possibly ...
3
votes
2answers
134 views

Is $\sqrt[3]{2}$ contained in $\mathbb{Q}(\zeta_n)$?

Is $\sqrt[3]{2}$ contained in $\mathbb{Q}(\zeta_n)$ for some $n$, where $\zeta_n=e^{2\pi i/n}$? I think the answer is no, but I can't give a full proof. Assume the contrary, we then have ...
3
votes
2answers
467 views

Purely inseparable extension

Let $F\subset K$ be an algebraic field extension. Is the set of all elements of $K$ that are purely inseparable over $F$ necessarily a subfield of $K$?
2
votes
3answers
56 views

Simple field extension inequality proof

Let $\alpha \in \mathbb{C}$ be algebraic over $\mathbb{Q}$ and $F\subseteq \mathbb{C}$ be a subfield. Prove that $[F(\alpha):F]\leqslant [\mathbb{Q}(\alpha):\mathbb{Q}]$. This looks like a problem ...
1
vote
2answers
57 views

Let $\alpha \in \overline{\Bbb Q}$ a root of $X^3+X+1\in\Bbb Q[X]$. Calculate the minimum polynomial of $\alpha^{-1}$ en $\alpha -1$.

Let $\alpha \in \overline{\Bbb Q}$ a root of $X^3+X+1\in\Bbb Q[X]$. Calculate the minimum polynomial of $\alpha^{-1}$ en $\alpha -1$. I don't really understand how to get started here. I know ...
1
vote
2answers
90 views

In characteristic $p$, the field extension $k(X,Y)$ over $k(X^p,Y^p)$ has degree $p^2$

Let $k$ be a field with characteristic $p>0$, $L=k(X,Y)$ be the field of rational fractions of two variables over $k$. Let $K=k(X^p,Y^p)$. Prove that $[L:K]=p^2$ Help me a hint to prove it. ...
1
vote
2answers
214 views

Roots of polynomials over finite fields

I've been trying to find the decomposition of $x^2-2$ to irreducible polynomials over $\mathbf{F}_5$ and $\mathbf{F}_7$. I know that for some $a$ in $\mathbf{F}_5$ (for example), $x-a$ divides $x^2-2$ ...
1
vote
1answer
66 views

Dimension Recovery of $S \subset P_n(F)$

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset? Added from answer posted by Trancot on 18 Apr ...
1
vote
1answer
56 views

Why $g(x^{p})=(g(x))^{p}$ in the reduction mod $p$?

In one of the proof in the book "Abstract Algebra'' by Dummit and Foote (Theorem 41, pg. 554) we have a monic polynomial $g(x)\in\mathbb{Z}[x]$, and the book claims that $g(x^{p})=(g(x))^{p}\mod p$ ...
1
vote
1answer
167 views

A question about splitting field [duplicate]

Possible Duplicate: Splitting field of $x^{n}-1$ over $\mathbb{Q}$ Let $A$ denote the splitting field of $x^n-1$ over $\mathbb{Q}$, so what is the dimension of $A$ as a linear space over ...
0
votes
1answer
564 views

Cyclotomic polynomial over a finite prime field [duplicate]

Possible Duplicate: Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})\[X\]$ Let $p$ be a prime number. Let $l$ be an odd prime number such that $l \neq p$. Let $X^l - 1 \in ...
0
votes
2answers
305 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, ...
18
votes
2answers
531 views

field generated by a set

Let $S$ be the set of real numbers which can be written in the form $ \sum_{n\geq0}{ \frac{\epsilon_{n}}{n!}}$ ,where ${\epsilon_n}^2=\epsilon_n$ and let $K$ be the field generated by $S$ , help me ...
21
votes
1answer
253 views

Is a field determined by its family of general linear groups?

Assume that $K,L$ are fields such that there is an isomorphism of groups $\mathrm{GL}_n(K) \cong \mathrm{GL}_n(L)$ for all $n \in \mathbb{N}$. Does it follow that $K \cong L$? I am also interested in ...
20
votes
2answers
526 views

What is the coproduct of fields, when it exists?

This is a slightly more advanced version of another question here. Let $\textbf{CRing}$ be the category of commutative rings with unit. Let $\textbf{Dom}$ be the category of integral domains – by ...