Rings and Fields¶
Midterm: 2025-05-07T10:00-03:00
Final exam: 2025-06-25T10:00-03:00
Make-up exam: 2025-07-02T10:00-03:00
Exercises.
P1 Q2.1¶
Define characteristic of a ring.
P1 Q2.2¶
A field with characteristic zero has a subfield isomorphic to the rationals.
P1 Q3.1¶
Let \(C\) be the ring of functions \(\mathbb{R}^{\mathbb{R}}\) with the pointwise operations addition and product. For each \(r\in\mathbb{R}\), let \(M(r)\) be the subset of \(C\) given by \(M(r)=\{f\in C:f(r)=0\}\).
Define the concept of maximal ideal.
P1 Q3.2¶
Show that \(M(r)\) is a maximal ideal.
P1 Q4.1¶
Define sum and product operations between formal power series rings.
P1 Q4.2¶
Let \(R\) be an integral domain. Prove that the ring of formal power series \(R[\![x]\!]\) is an integral domain.
P1 Q5.1¶
Define Euclidean domain.
P1 Q5.2¶
Let \(R\) be a Euclidean domain equipped with the Euclidean function \(\nu :R\setminus \{0\}\rightarrow\mathbb{N}\) and \(a\in R\) nonzero. Let \(m\) be the least integer taken by \(\nu\). Prove that \(a\in R\setminus \{0\}\) is invertible iff \(\nu (a)=m\).
P2 Q1.3 E10.7¶
Let \(F/E\) be a field extension, and \(u\in F\). Prove that, if the degree of \(u\) over \(E\) is finite and odd, then \(E(u)=E(u^2)\).
P2 Q2.1¶
Eisenstein’s irreducibility criterion.
P2 Q2.4¶
Let \(p\in\mathbb{Z}\) be a positive prime. For which integers \(n\geq 1\) is the real number \(\sqrt[n]{p}\) constructible?
P2 Q3.2¶
Let \(R\) be a UFD and \(a_0,...,a_n\in R\) not all of which are zero. Prove that \(1\in GCD(a_0,...,a_n)\) iff for all \(p\in R\) irreducible, there exists \(i\in\{0,...,n\}\) such that \(p\nmid a_i\).
Hint: For the forward direction, use contraposition. For the reverse direction, let \(d\) be a common divisor of \(a_0,...,a_n\), then \(d=u\cdot p_1^{e_1}\cdot p_2^{e_2}\dotsb p_k^{e_k}\).
P2 Q4.1¶
Define algebraic element over smaller field. Define algebraic extension.
P2 Q4.2¶
If \(F/E\) is an algebraic field extension and \(D\) is a subring of \(F\) containing \(E\), then \(D\) is a field.
Subring of an algebraic field extension is a subfield [MathSE].
P2 Q4.3¶
For the previous item, provide an example that shows the necessity of the algebraicness hypothesis.
Hint: Consider the field extension \(\mathbb{Q}(\pi ,\sqrt{2})\), and the subring \(\mathbb{Q}[\pi ,\sqrt{2}]=\{a+b\sqrt{2} +c\pi +d\pi\sqrt{2} :a,b,c,d\in\mathbb{Q} \}\)
P2 Q4.4¶
Let \(L\) be a subfield of \(\mathbb{R}\) and \(a\in\mathbb{R}\) non-negative. Prove that \(a\) is algebraic over \(L\) iff \(\sqrt{a}\) is algebraic over \(L\).
PSub Q1.2¶
Let \(A\) be a ring and \(S\) a non-empty collection of subrings of \(A\) such that for all \(B,C\in S\), there exists \(D\in S\) such that \(B\cup C \subset D\). Prove that \(\bigcup_{B\in S}B\) is a subring of \(A\).
Hint: Apply the subring test.
PSub Q2.2¶
PSub Q3.1¶
Let \((A_i)\) be a family of rings. Define the product of rings \(\prod_{i\in I} A_i\).
PSub Q3.3¶
Prove or give a counterexample. The product of two integral domains is an integral domain.
E5.1 PSub Q3.4¶
List all elements of \(\mathbb{Z}_{12}\) that are zero divisors.
E9.6 PSub Q3.2¶
Consider \(R=\mathbb{Z}_4^{\mathbb{N}}\) with the structure of a product of rings. Let \(a\in R\) be given by \(a(n)=2\) for all \(n\in \mathbb{N}\). Prove that the polynomial \(ax\in R[x]\) has infinite roots.
PSub Q4.1¶
Define finitely generated field extension.
E10.5 PSub Q4.2¶
Give an example of a finitely generated field extension that doesn’t have finite degree. Explain.
Field extension that is finitely generated but not finite dimensional [MathSE].
E10.8. Same as E10.9.
E10.9. Let \(F/E\) be an algebraic extension, and \(D\) a subring of \(F\) containing \(E\). Prove that \(D\) is a field. Provide an example that shows that the algebraicness condition is necessary.
E10.10 PSub Q4.3, Q4.4¶
Let \(F/E\) be a field extension, and \(u,v \in F\) elements of degree \(m,n < \infty\) over \(E\), respectively. Prove that \([E(u,v):E]\leq mn\). Prove also that, if \(m,n\) are coprime, then \([E(u,v):E]=mn\).
Extension fields of coprime degrees [MathSE].
E10.11. Prove that the subfields \(\mathbb{Q} (i)\) , \(\mathbb{Q} (\sqrt{2})\) of \(\mathbb{C}\) are isomorphic as \(\mathbb{Q}\)-vector spaces, but not isomorphic as fields.
Fields isomorphic as vector spaces but not as fields [MathSE].
Is \(\mathbb{Q}\sqrt{2} \cong \mathbb{Q}\sqrt{3} ?\) [MathSE Link].
Why \(\mathbb{Q}\sqrt{2}\) and \(\mathbb{Q}\sqrt{3}\) are isomorphic as \(\mathbb{Q}\)-vector spaces but not as fields? [MathSE link].
Examples of fields that are isomorphic, infinite, and not equal.
Comment: Page 241 of Hungerford, according to the MathSE post “\(\mathbb{C}\) and \(\mathbb{Q}\) are isomorphic as vector spaces but not as fields” [Link] (problem statement has a typo).
E10.19 PSub Q4.5¶
If \(F\) is a finite field, then the cardinality of \(F\) is \(p^n\) for some prime \(p\) and some \(n\in\mathbb{N}\).