Field Extensions

Let \(F\) and \(E = F(\alpha )\) be two fields such that \([E:F]\) is odd. Prove that \(E = F(\alpha ^2)\).

Let \(f(x)\) be an irreducible polynomial in \(\mathbb {Q}[x]\), with splitting field \(K\). Suppose \(\operatorname {Gal}(K/\mathbb {Q})\) is abelian. Prove or disprove: \(K = \mathbb {Q}(\alpha )\) for each root \(\alpha \) of \(f(x)\) in \(K\).

Let \(F\) be a field of characteristic zero and let \(E/F\) be a finite Galois extension. Suppose that \(E = F[\alpha ]\). Suppose that there exists \(\sigma \in G = \operatorname {Gal}(E/F)\) with \(\sigma (\alpha )=-\alpha \). Prove that \(E \ne F[\alpha ^2]\).

Determine whether the following two fields \(K\) and \(L\) are isomorphic; if they are isomorphic then construct an explicit isomorphism between them.

Let \(2^{1/4}\) be the unique positive real fourth root of of 2 in \(\mathbb {R}\). Show that \(\mathbb {Q}(2^{1/4})/\mathbb {Q}\) is not a Galois extension.

With \(\alpha =\sqrt {6+\sqrt {11}}\), prove that the extension \(\mathbb {Q}[\alpha ]/\mathbb {Q}\) is Galois and compute its Galois group.

Describe a primitive generator for the degree two extension of \(\mathbb {Q}\) inside the extension \(\mathbb {Q}(\zeta )\), where \(\zeta :=e^{2\pi i/7}\). Justify that the extension you identify is of degree two.

Prove that \(\mathbb {Q}(\sqrt [3]{2})\) is not a subfield of any cyclotomic field over \(\mathbb {Q}\).

Let \(A=\mathbb {F}_3[x]\), i.e., \(A\) is a polynomial ring in one variable over the field with 3 elements. Suppose \(M\) and \(N\) are finitely generated \(A\)-modules such that \[ M\oplus \frac {A}{x^3+1} \cong N\oplus \frac {A}{x+1}\oplus \frac {A}{x+1}\oplus \frac {A}{x+1}. \] Are \(M\) and \(N\) isomorphic?

Determine the extension degree of the splitting field of \(x^7-1\) over \(\mathbb {F}_{11}\).

Let \(M/K\) be a Galois extension of degree 700. Prove that there exists an intermediate field \(L\) with \([L:K]=28\). Is \(L/K\) necessarily Galois?

Find a Galois extension of \(\mathbb {Q}\) with Galois group isomorphic to \(\mathbb {Z}/3\times \mathbb {Z}/3\).

Let \(F\subset L\) a extension of fields of degree 4. Prove that there are no more than 3 fields proper intermediate subfields \(K\); namely, such that \(F\subset K\subset L\).

Let \(F\) be a field and \(n\) a positive integer such that \(F\) has no nontrivial field extensions of degree less than \(n\). Let \(L=F[\alpha ]\) be an extension such that \(\alpha ^n\) is in \(F\). Prove that each element of \(L\) is a product of elements of the form \(a\alpha +b\), with \(a,b\) in \(F\).

Let \(\alpha =2^{1/5}+5^{1/2}\). Determine the extension degree of \(\alpha \) over \(\mathbb {Q}\).

Let \(k\) be a field and \(k(x)\) the field of rational functions over \(k\), in the indeterminate \(x\). Consider the automorphisms \(\sigma \) and \(\tau \) of \(k(x)/k\) defined by \[ \sigma (x)=\frac {1}{1-x}\qquad \text {and}\qquad \tau (x)=\frac {1}{x}. \]

1. Prove that the subgroup \(G=\langle \sigma ,\tau \rangle \) of \(\operatorname {Aut}(k(x)/k)\) is isomorphic to \(S_3\).
2. Prove that the fixed subfield under the action of \(G\) equals \(k(t)\), where \[ t=\frac {(x^2-x+1)^3}{x^2(x-1)^2}. \]

Let \(n\) a positive integer, and let \(\mathbf {x}:=x_1,\ldots ,x_n\) be indeterminates over \(\mathbb {Q}\). For each positive integer \(i\) set \[ \alpha _i=x_1^i+\cdots +x_n^i. \]

1. Clearly \(\mathbb {Q}(\alpha _1,\ldots ,\alpha _n)\subseteq \mathbb {Q}(s_1,\ldots ,s_n)\), the field of symmetric functions in the \(\mathbf {x}\). What is the degree of this extension?
2. Prove the characteristic polynomial of an \(n\times n\) matrix \(A\) over \(\mathbb {Q}\) is completely determined by the elements \(\operatorname {trace}(A),\ldots ,\operatorname {trace}(A^n)\).

Let \(a,b,c,d\) integers such that \[ \det (A)\ne 0\qquad \text {where}\qquad A=\begin {pmatrix} a & b\\ c & d \end {pmatrix}. \] Let \(x\) and \(y\) be indeterminates, and set \(L=\mathbb {C}(x,y)\) and \(K=\mathbb {C}(x^a y^b,x^c y^d)\).

1. Prove that \(L\) is a finite extension of \(K\) and that \((L:K)=|\det (A)|\).
2. Show that \(L/K\) is Galois, and compute the Galois group.

Let \(L/K\) be a Galois extension of fields of characteristic zero. Suppose \(f(x)\) is an irreducible polynomial of degree 5 in \(K[x]\). If \(f(x)\) has no roots in \(L\), prove that it is irreducible in \(L[x]\).

Let \(E\) be the splitting field of \(f(x)=x^{14}+1\) over \(\mathbb {F}_2\), and let \(K\) be the splitting field \(g(x)=x^{21}+1\) over \(\mathbb {F}_2\). Prove that \(K\) contains an isomorphic copy of \(E\), and compute the extension degree of \(K\) over \(E\).

Let \(K\) be a field of characteristic \(p>0\), and let \(t\) be an indeterminate. Consider the automorphism \(\sigma \in \operatorname {Aut}_K K(t)\) with \(\sigma (t)=t+1\). Determine the subfield of \(K(t)\) that is fixed by \(\sigma \).

Let \(\alpha \) be a complex root of \(x^6+x^3+1\). Determine all field homomorphisms \(\mathbb {Q}(\alpha )\longrightarrow \mathbb {C}\).

Let \(M\) be an \(n\times n\) matrix with entries from a field \(\mathbb {F}\) such that \(M^3=I\). Is \(M\) necessarily diagonalizable if (a) \(\mathbb {F}=\mathbb {Q}(\sqrt {3})\), (b) \(\mathbb {F}=\mathbb {Q}(i)\), (c) \(\mathbb {F}=\mathbb {F}_3\)?

Let \(\mathbb {F}\) be a field with \(\mathbb {Q}\subset \mathbb {F}\subset \mathbb {C}\) such that \([\mathbb {F}:\mathbb {Q}]\) is odd.

1. If \(\mathbb {F}/\mathbb {Q}\) is Galois, prove that \(\mathbb {F}\) is contained in \(\mathbb {R}\).
2. Find an extension with \([\mathbb {F}:\mathbb {Q}]=3\) such that \(\mathbb {F}\) is not contained in \(\mathbb {R}\).

Let \(\alpha \) be a complex root of \(x^6+3\). Set \(K=\mathbb {Q}(\alpha )\).

1. Prove that \(K\) contains a primitive 6-th root of unity.
2. Compute the Galois group of \(K\) over \(\mathbb {Q}\).

What is the Galois group of \(x^4-5\) over:

1. \(\mathbb {Q}\);
2. \(\mathbb {Q}(\sqrt {5})\);
3. \(\mathbb {Q}(i)\).

Show that the extension \(\mathbb {Q}(\sqrt {2},\sqrt {3})\) is Galois over \(\mathbb {Q}\), and determine the Galois group.

Let \(F\) be a field that contains a primitive \(n\)th root of unity. Show that if \(a\) is an element of \(F\) and the field \(E\) is obtained from \(F\) by adjoining an \(n\)th root of \(a\), then \(E\) is a Galois extension of \(F\) with cyclic Galois group.

Let \(E\subset F\) be a finite Galois extension of fields. Suppose that there is an element \(\alpha \) in \(F\) such that \(\alpha \notin E\) and such that \(\alpha \) is in every proper extension of \(E\) contained in \(F\). Show that the Galois group of \(F\) over \(E\) is cyclic of prime power order.

Show that there exists a Galois extension of the field \(\mathbb {Q}\) of rational numbers with Galois group \(\mathbb {Z}/5\mathbb {Z}\).

Let \(K\) be a subfield of a field \(L\). If \(\sigma :L\longrightarrow L\) is a homomorphism that fixes all elements of \(K\), is \(\sigma \) necessarily an automorphism of \(L\)? Justify your answer.

Let \(\omega =e^{2\pi i/3}\). Let \(\sigma \) and \(\tau \) be automorphisms of \(\mathbb {C}(x)\) which fix \(\mathbb {C}\) and satisfy \[ \sigma (x)=\omega x,\qquad \tau (x)=1/x. \] Prove that the group \(\langle \sigma ,\tau \rangle \) is isomorphic to the symmetric group \(S_3\), and determine the subfield of \(\mathbb {C}(x)\) that is fixed by this group.

Find a real number \(\alpha \) such that \(\mathbb {Q}(\alpha )\) is a Galois extension of \(\mathbb {Q}\) with the Galois group \(\mathbb {Z}/5\mathbb {Z}\).

Let \(F\) be a field with \(\mathbb {Q}\subset F\subset \mathbb {C}\), where \(F/\mathbb {Q}\) is a finite Galois extension. Let \(\alpha \in F\) and let \(f(X)\in \mathbb {Q}[X]\) be its minimal monic polynomial. Assume that the absolute value \(|\alpha |\) is 1, and that \(\operatorname {Gal}(F/\mathbb {Q})\) is abelian.

1. Show that \(F\) is closed under complex conjugation.
2. Prove that \(|\beta |=1\) for every complex root \(\beta \) of \(f(X)\).
3. Writing \(f(X)=X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0\), show that \(|a_i|\le 2^n\) for all \(i\) with \(0\le i<n\).
4. Prove that \(F\) contains only finitely many algebraic integers having absolute value 1 and deduce that each of these is a root of unity.

Determine the group of units in the ring \(\mathbb {Q}[x]/((x^2+1)^2)\).

Let \(K\) and \(F\) be fields such that \(K\) is a finite extension of \(F\). Assume that there are only finitely many subfields \(L\) such that \(F\subset L\subset K\). Show that \(K\) is a primitive extension of \(F\) (i.e. \(K=F(\theta )\) for some \(\theta \in K\)).

Find a galois extension \(K\) of \(\mathbb {Q}\) with Galois group \(\mathbb {Z}/5\mathbb {Z}\oplus \mathbb {Z}/5\mathbb {Z}\).

Let \(F\) be a finite field, and suppose that the subfield of \(F\) generated by \(\{x^3\mid x\in F\}\) is different from \(F\). Show that the cardinality of \(F\) is 4.

Let \(E\) be an extension of the field \(F\) and let \(\alpha \) be an element of \(E\) which is algebraic over \(F\) of degree \(n\).

1. If \(n\) is prime to 2, prove that \(F(\alpha ^2)=F(\alpha )\).
2. Give an example in which \(n\) is prime to 3 but \(F(\alpha ^3)\ne F(\alpha )\).

1. Suppose \(K\) is a field of prime order and \(E\) is the algebraic closure of \(K\). If \(F\) is any subfield of \(E\), prove that the multiplicative group of \(F\) is periodic, i.e., every element of the multiplicative group has finite order.
2. Suppose \(F\) is a field whose multiplicative group is periodic. Prove that for some prime \(p\), \(F\) is isomorphic to a subfield of the algebraic closure of the field of order \(p\).

Let \(E\subset F\) be an extension of finite fields. Let \(q\) be the number of elements of \(E\), and let \(\phi \colon F\to F\) be defined by \(\phi (a)=a^q\). Prove that \(F\) is a Galois extension of \(E\) and that \(\phi \) generates its Galois group.

Let \(E\subset F\) be an extension of finite fields. Prove that the trace map from \(F\) to \(E\) is surjective.

Let \(E\) be a finite separable extension of a field \(K\) and let \(F\) be any finite field extension of \(K\). Show that \(E\otimes _K F\) is a finite product of fields.

Show that there exists a Galois extension of the rational numbers with Galois group \(\mathbb {Z}/7\mathbb {Z}\).

Suppose \(F\) is a field and \(K=F(\alpha )\) is a finite Galois extension of \(F\). Assume that the Galois group of \(K\) over \(F\) is cyclic generated by an automorphism \(\sigma \) where \(\alpha ^\sigma =\alpha +1\). Prove that \(F\) has characteristic \(p\ne 0\) and that \(\alpha ^p-\alpha \in F\).

Let \(F\) be a field, and let \(f(X)\) be a polynomial in \(F[X]\). Show that \(f(X)\) has a multiple root in some extension of \(F\) if and only if \(f(X)\) and its derivative \(f'(X)\) are not relatively prime.

Show that if \(\alpha \) and \(\beta \) are elements of a field \(E\) which are algebraic over a subfield \(F\), then \(\alpha +\beta \) and \(\alpha \beta \) are algebraic over \(F\).

Let \(E\) be a normal extension of \(\mathbb {Q}\) contained in \(\overline {\mathbb {Q}}\) (the algebraic closure of \(\mathbb {Q}\)), and let \(K\) be an extension of \(\mathbb {Q}\) contained in \(E\). Let \(K'\) be a subfield of \(\overline {\mathbb {Q}}\) which is isomorphic to \(K\). Show that \(K'\) is contained in \(E\) and the Galois groups of \(E/K\) and \(E/K'\) are conjugate in the Galois group of \(E/\mathbb {Q}\).

Let \(K\) be an arbitrary field, and let \(L\) be an extension of \(K\) obtained by adjoining a root of the polynomial \(X^n-1\) for some integer \(n\ge 1\). Show that \(L\) is Galois over \(K\) with abelian Galois group.

Let \(K/k\) be a Galois extension with Galois group \(G\). Show that every 1-cycle on \(G\) with values in \(K^*\) is a 1-coboundary.

If \(H\) is a finite group, prove that there exists a field \(k\) and a Galois extension of \(k\) with Galois group \(H\).

Let \(E\) be an extension of the field \(F\) and let \(\alpha \) be an element of \(E\) which is algebraic over \(F\) of degree \(n\).

1. If \(n\) is prime to 2, prove that \(F(\alpha ^2)=F(\alpha )\).
2. Give an example in which \(n\) is prime to 3 but \(F(\alpha ^3)\ne F(\alpha )\).

Let \(K\subseteq L\) be finite fields. Prove that the extension \(L\) of \(K\) is Galois with cyclic Galois group.

Let \(E\) be a finite extension of the rationals. Show that the number of roots of unity in \(E\) is finite.

Suppose \(F\) is a field of characteristic 0 with the property that every polynomial of odd degree over \(F\) has at least one root in \(F\). Assume that \(E\) is an extension of degree 2 over \(F\) with the property that for each \(\alpha \in E\), the polynomial \(x^2-\alpha \) has a root in \(E\). Prove that \(E\) is algebraically closed.

Let \(F\) be the field of integers modulo 7. Determine which finite extensions of \(F\) contain exactly four 4th roots of 1.