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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3
votes
1answer
341 views

Dimension of the splitting field of $f = x^3 − x + 1$

Let $L_f$ be the splitting field of the irreduicble polynomial $f = x^3 − x + 1$ over $\Bbb{Q}[x]$. I want to determine $\operatorname{dim}_{\Bbb{Q}}L_f$. $f$ has three roots in its splitting field ...
2
votes
2answers
471 views

Compositum of abelian Galois extensions is also?

Suppose I have a field $k$ and two extensions $K/k$ and $L/k$ which are both abelian Galois extensions of $k$. Then (assuming $K$ and $L$ are both contained in some bigger field) is the compositum ...
3
votes
3answers
341 views

Algebraic Elements and Fields of Quotients

The algebraic elements of $\mathbb{R}$ are those elements which are roots of nonzero polynomials with coefficients in $\mathbb{Q}$. In fact, by multiplying through by denominators, we can even take ...
1
vote
0answers
77 views

Irreduciblity of the polynomial $x^{p^n}-x+1$ [duplicate]

Possible Duplicate: Why $x^{p^n}-x+1$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$ What are the values of $n$ for which the polynomial $$f(x):=x^{p^n}-x+1$$is irreducible ...
2
votes
2answers
206 views

Splitting field and dimension of irreducible polynomials

Given a field extension $L/K$, $\alpha, \beta \in L$ and $f,g \in K[x]$ irreducible polynomials with $f(\alpha)=g(\beta)=0$. Then $$ \operatorname{dim}_K(K(\alpha,\beta)) = \deg(f) \cdot ...
2
votes
4answers
264 views

Dimension of a splitting field

Given a field $F$ and a polynomial $P \in F[x]$ such that $P$ is irreducible over $F$. Let $L_P$ be the splitting field of $P$ and $F$. Does $\operatorname{dim}_F({L_F}) = \deg(P)$ hold? If looking ...
2
votes
1answer
104 views

totally real number field generated by square root of an algebraic integer

Let $d$ be a real positive algebraic integer of degree $3$ or $5$. Assume that $\mathbb{Q}(d)$ and $\mathbb{Q}(\sqrt{d})$ are totally real number fields. Is there a possible $d$ which makes that ...
1
vote
2answers
242 views

Are these fields isomorphic?

Are the fields $\mathbb{Q}(i)$ and $\mathbb{Q}(2i)$ isomorphic? I'm confused since they seem to be equal as sets but $\mathbb{Q}(i)\cong \mathbb{Q}[X]/(X^2+1)$ but $\mathbb{Q}(2i)\cong ...
1
vote
2answers
484 views

How does a permutation of roots induce an automorphism on a splitting field?

I'll be learning Galois theory for the first time later this year and wanted to clear up something that was puzzling me. If f(x) = $\sum {a_ix^{i}}$ is a polynomial which is irreducible over ...
1
vote
1answer
235 views

What is the purpose of the characteristic exponent?

I just came across the term "characteristic exponent" of a field $\Bbbk$. Apparently, it is equal to $1$ if $\DeclareMathOperator{\c}{char}\c(\Bbbk)=0$ and it is equal to $p=\c(\Bbbk)$ otherwise. ...
6
votes
3answers
903 views

Field with natural numbers

To make sure that we are talking about the same, I would like to post the relevant definitions I know first. Definitions: A pair $(G, +)$ where $G$ is a set and $+: G \times G \rightarrow G$ is ...
1
vote
1answer
116 views

Field theory, extensions

HiAll, I am stuck with this problem: (a) Let $K$ be a field such that characteristic of $K$ is not 2. Prove that any extension $L$ of $K$ with $K\subset L$, and $[L:K]=2$ has the form $L=F(\beta)$ ...
2
votes
1answer
2k views

Proving a polynomial irreducible over finite field [duplicate]

Possible Duplicate: How do I prove that $x^p-x+a$ is irreducible in a field of $p$ elements when $a\neq 0$? How to prove that $x^p - x - 1$ is irreducible over $F_{p}[x]$. I thought ...
0
votes
1answer
80 views

Field extension question, basis, cannot be expressed as linear combination?

Suppose I have a field extension K of F with basis $\{1,\beta\}$, $\beta\in K^*/F^*$. How do I show that $\beta^2$ cannot be written as $c_1+c_2\beta$, where $c_1,c_2\in F, c_2\ne 0$ unless $\beta^2 ...
1
vote
1answer
209 views

Notation for finite fields

What is the meaning of the following: If $q(x)$ is irreducible polynomial of degree $d$ and $d$ divides $n$, then $q(x)$ divides $x^{p^n}-x$. Let $F = F_p[x]/(q(x)) = F_p[\alpha]$ where ...
3
votes
1answer
367 views

Non-Archimedean Fields

Are there non-Archimedean fields without associated valuation or being a non-archimedan field implies it is a valuation field? I understand that a non-Archimedean field is a field which does not ...
4
votes
2answers
280 views

How to determine if this is a field?

A finite subring $R$ of a field $V$ contains $1$ (so $1$ is an element of $R$). The question is: True or False: The ring $R$ must be a field. I thought that if $R$ was a field it had to be a finite ...
2
votes
1answer
248 views

Primitive Element for Field Extension of Rational Functions over Symmetric Rational Functions

A rational function $f$ in $n$ variables is a ratio of $2$ polynomials, $$f(x_1,...x_n) = \frac{p(x_1,...x_n)}{q(x_1,...x_n)}$$ where $q$ is not identically $0$. The function is called symmetric if ...
10
votes
5answers
2k views

Can you construct a field with 6 elements? [duplicate]

Possible Duplicate: Is there anything like GF(6)? Could someone tell me if you can build a field with 6 elements.
3
votes
1answer
200 views

Commutative Algebra - Polynomial Rings

Let $Z$ be the ring of integers, $p$ a prime and $F_p = Z/pZ$ the field with $p$ elements. Let $x$ be an indeterminate. Set $R_1 = F_p[x]/(x^2-2)$, $R_2 = F_p[x]/(x^2-3)$. Determine whether the rings ...
7
votes
2answers
280 views

Splitting field of $x^{13}+1$ over $\mathbb{Q}$

I want to know how to calculate the degree of the field extension $[K:Q]$ where $K$ is the splitting field of $x^{13}+1$ over $\mathbb{Q}$. I'm new to this area and this is not really covered in my ...
2
votes
1answer
69 views

Extending $\phi: A \rightarrow \Omega$ to $A[x] \rightarrow \Omega$ where $A$ is integral domain and $x$ transcendental over $A$

Let $A \subseteq B$ be integral domains and let $\phi:A \rightarrow \Omega$ be a homomorphism of $A$ into the infinite algebraically closed field $\Omega$. Let $x \in B$ and suppose that $x$ is ...
5
votes
1answer
178 views

Finding the splitting field of $f(x)$

I'm trying to learn the theory of splitting fields. So I went through this example on an old exam: Find the splitting field $K$ of $f(x)$ over $\mathbb{Q}$ for $f(x)=x^6-9$ $x^6-9=(x^3-3)(x^3+3)$ and ...
8
votes
1answer
238 views

Generator of the multiplicative groups of units in $\mathbb{Z_5[x]}/(x^{2}+3x+3)$

I would like to find a generator of the multiplicative groups of units in $\mathbb{Z_5[x]}/(x^{2}+3x+3)$. This is a field since $x^{2}+3x+3$ is irreducible, so every coset with $bx+a\not=0$ as a ...
7
votes
1answer
279 views

Extension of Homomorphisms (Lang, Atiyah and McDonald)

Let $A$ be a subring of a field $K$, and suppose that $A$ is a local ring with maximal ideal $\mathfrak{m}$. Let $x \in K, \, x \neq 0$. Let $\phi: A \rightarrow L$ be a homomorphism of $A$ into the ...
0
votes
2answers
108 views

Extension of an embedding to a field extension by a transcendental element

Let $K$ be a subfield of a field $K'$ and suppose we have an embedding $\phi:K \rightarrow L$ of $K$ into an algebraically close field $L$. Let $x \in K'$. If $x$ is algebraic over $K$, then we can ...
3
votes
5answers
245 views

Find an ideal $I$ of $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/{I}$ is a field

Find an ideal $I$ of $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/{I}$ is a field. How can one justify the answer in the shortest number of lines?
1
vote
1answer
86 views

Automorphism and Stabilizer

Let $K \leq E \leq F$ be fields such that $[E:K]=n$. Let $Aut_K F$ act on the set $S$ of intermediate fields where $\sigma(I)$ gives the action of $\sigma \in Aut_K F$ on an intermediate field $I$. ...
3
votes
4answers
194 views

Nonzero elements of splitting field

Let $F$ be a splitting field of $x^{p^{n}} - x \in \mathbb{Z}_p[x]$. How is it that the nonzero elements multiply to $-1$ and sum to $0$? I don't get how we get that result.
41
votes
2answers
3k views

What kind of work do modern day algebraists do?

Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
3
votes
1answer
120 views

Trace and Norm maps on differential extensions

I'm working through a proof which is rather algebraic, and my abstract algebra is probably only basic to intermediate. I have a differential extension $E/K$ of a differential field $K$, and the proof ...
2
votes
1answer
96 views

Latin squares of even order with all cells only participating in one subsquare.

For even ordered Latin squares, we can create squares in which every cell participates in a $2\times 2$ subsquare by using a simple circulant as for $n=6$ below: ...
2
votes
2answers
211 views

Irreducible polynomials with integer coefficients over Q

Suppose p(x) is an irreducible polynomial over Q of degree n, with integer coefficients. If p(x) has two roots r1 and r2 satisfying r1r2 = 5, prove that n is even. Attempt at solution: Because the ...
2
votes
1answer
94 views

Proof about Field conjugation isomorphisms

I'm having an awful time making sense of a proof and I was hoping someone could help. Theorem: Let $\alpha$ and $\beta$ be algebraic over a field $F$ with $deg(\alpha, F) = n$, as elements of a ...
0
votes
1answer
75 views

Maximal Separable Subextension is Finite?

Consider the following statement: "Let $L/K$ be an algebraic field extension. Then the maximal separable sub-extension is finite." Here is what seems to be a proof: "Let $M/K$, $K \subset M \subset ...
1
vote
2answers
436 views

Number of field homomorphisms from an extension field of $\mathbb Q$ to $\mathbb C$

Take $\mathbb{Q}$ $\subset$ $K$ $\subset$ $\mathbb{C}$ with $[K:\mathbb{Q}]$ finite. How would you show that the number of field homomorphisms from $K$ to $\mathbb{C}$ is equal to $[K: ...
8
votes
2answers
920 views

How can one express $\sqrt{2+\sqrt{2}}$ without using the square root of a square root?

I was trying to review some analysis, and came across problem 3 from page 78 of Walter Rudin's Principles of Mathematical Analysis. As part of the problem, I wanted to try to write ...
3
votes
1answer
156 views

Relation of compositum of fields

Let $E/k$ be a finite field extension, $\operatorname{char}(k)=p>0$. Suppose that $E^p k = E$. Is it then true that $E^{p^n}k = E$ for any positive integer $n$? If yes, why? Thanks.
2
votes
3answers
111 views

Uniqueness of prime-power fields

I'm still stuck on the proof of the following theorem. I've asked two questions so far to get to where I am even at this point. Theorem: Let $p$ be a prime and let $n\in\mathbb{Z}^{+}$. If $E$ and ...
1
vote
3answers
98 views

Question about algebraic field extensions

If I have a subfield $F$ of a field $E$, and an algebraic (over $F$) $\alpha\in E$, I can form $F(\alpha)$ which is isomorphic to $F[x]/\langle f(x)\rangle$ for $f(x) = irr(\alpha, F)$. That is, ...
0
votes
1answer
83 views

Can't follow a proof involving Prime-Power Fields

Theorem: Let $p$ be a prime and let $n\in\mathbb{Z}^{+}$. If $E$ and $E'$ are fields of order $p^{n}$, then $E\cong E'$. Proof: Both $E$ And $E'$ have $\mathbb{Z}_{p}$ as prime fields (up to ...
3
votes
1answer
133 views

Is an automorphism of a normal extension determined by its image of the maximal separable sub extension?

Let $L / K$ be a normal, algebraic field extension. Suppose that the maximal separable sub- extension $M/K$ is finite, $K \subseteq M \subseteq L$. By the primitive element theorem, $M=K(x)$ for some ...
0
votes
2answers
112 views

Question about isomorphism between a ideal and a polynomial ring

Sorry for my ignorance, my question is: Let be $F[X]$ a polynomial quotient ring, where $F$ is a finite field with characteristic 2. Are there any ideal, $I$, such that $I$ is isomorphic to $F[X]$?.
1
vote
1answer
92 views

Question about a corollary about Finite Fields

Definition: A field extension $E$ of $F$ is of degree $n$ (and is called a finite field extension) if $E$ is an $n$-dimensional vector space over $F$. Theorem: Let $E$ be a degree $n$ finite ...
4
votes
1answer
130 views

Unramified extension is normal if it has normal residue class extension

Let $K/F$ be an unramified extension such that $\rho_K / \rho_F$ (the corresponding extension of residue classes) is normal. Prove $K/F$ is normal. I guess I need to do some polynomial lifting, but ...
0
votes
1answer
150 views

What is a valuation associated to an ordering on a field?

If $(K,\leq)$ is a totally ordered field with $P\!=\!\{\alpha\!\in\!K;\, 0\!\leq\!\alpha\}$, how is the valuation associated to $P$ defined? I was searching through Prestel & Delzell's Positive ...
4
votes
1answer
331 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 ...
2
votes
2answers
155 views

Is a polynomial solvable by roots iff every irreducible factor is?

Let $F$ be a field, I asked myself if $p(x)\in F[x]$ is solvable by radicals iff every irreducible factor is solvable by radicals. My thoughts: If every irreducible factor is solvable by roots then ...
3
votes
1answer
285 views

Irreducible Polynomials in Finite Fields

I'm reading through some notes online concerning finite fields, and attempting to come up with a proof that all finite fields of the same size are isomorphic. But I'm getting stuck at a certain point, ...
0
votes
1answer
53 views

Basic question about fractions

I'm solving some exercises about fields and am trying to find the inverse for $a_1 + \sqrt{2}b_1$, i.e. $\frac{1}{a_1 + \sqrt{2}b_1}$. This means I need to split the fraction into something of the ...