Mathematics 894 Take Home Exam April 08

The questions have equal value. You may attempt all questions if you wish. Your best five answers will be counted. There are three pages to this exam. You may freely use the text Abstract Algebra, Third Edition, by Dummit and Foote, and Algebra, Third Edition, by Lang. You should not consult other class members, or anyone else (except me in the case of a suspected typo or uncertainly about what a question menas), while working on this exam. It is due at 4PM on April 22 in my office.

1. (a) Prove that the polynomials f(X) = X^{3} + X + 1 and g(X) = X^{3} + X^{2} + 1

are irreducible over 𝔽_{2}.

(b) Write X^{8} − X as the product of irreducible polynomials over 𝔽_{2}.

(c) Explain why 𝔽_{2}[X]/(f) and 𝔽_{2}[X]/(g) are isomorphic fields.

(d) Give an explicit isomorphism between the fields 𝔽_{2}[X]/(f) and 𝔽_{2}[X]/(g).

(e) Let F = 𝔽_{2}[X]/(g) (one of the fields from part (c)). Explain why F is the splitting field of Y^{8} − Y. Elements of F can be represented uniquely by polynomials of the form a_{0} + a_{1}X + a_{2}X^{2} where a_{i} ∈ 𝔽_{2} are represented by 0 or 1. Using this representation for elements of F find the roots of each of the irreducible factors of Y^{8} − Y given in part (b). (Since X is being used to represent elements of F, I am using Y as an indeterminate over F in this part of the question.)

2. Let R be any commutative ring. Let R^{n} be the direct sum of n copies of R (with elements written as row vectors). (You may assume that R is an integral domain if that helps you with this question).

(a) v = (a_{1}, …, a_{n}) ∈ R^{n} is defined to be unimodular if the ideal in R generated by a_{1}, … a_{n} is R. Prove that Rv is a direct summand of R^{n} (i.e. R^{n} = Rv⊕P for some R-submodule P of R^{n}) if and only if v is unimodular.

(b) Prove that if v is unimodular then P (as in (a)) is uniquely determined up to isomorphism. (i.e. if P ⊕ Rv = Q ⊕ Rv = R^{n} for different submodules P, Q of ℝ^{n}) then we must have P ≅ Q).

(c) Prove that if P is free we must have P ≅ R^{n−1}.

(d) Show that P is free if and only if v is the first row of an invertible n × n matrix with entries in R.

(e) Let R = ℝ[x, y, z]/(x^{2} + y^{2} + z^{2} − 1). Show that v = (x, y, z) is unimodular.

(f) It can be shown that, for R as in (d), P is not free. (Don’t try to prove this!). However if we let R = ℂ[x, y, z]/(x^{2} + y^{2} + z^{2} − 1) show that P is free.

3. Let F be an infinite field and K a finite extension of F. Assume that K/F a Galois extension.

(a) Explain why there are only a finite number of intermediate fields F ⊆ L ⊆ K.

(b) Assume that K = F (a_{0}, …, a_{n}) and for t ∈ F let

γ_{t} = a_{0} + a_{1}t + ⋯ a_{n}t^{n}

Explain why there must be an intermediate field L such that L = F (γ_{t}) for an infinite number of values of t.

(c) Explain why L must equal K.

(d) Part (c) shows that if K is any Galois extension of F then there is an element γ such that K = F (γ) (γ is called a primitive element of K over F). Show that any finite separable extension of F has a primitive element.

(e) Let K = F (γ) be any finite separable extension (not necessarily Galois). Then K ≅ F [x]/(f(x)) where f is the minimal polynomial of γ over F. Prove that there is a one-to-one correspondence between the roots of f and the embeddings σ : K → \overline{F} over F (i.e. σ|_{F} is the canonical inclusion F → \overline{F}).

(f) Let n = [K : F]. Let σ_{1}, …, σ_{n} be the n different embeddings of K into \overline{F} obtained in part (e). Show that there is an isomorphims of rings

ϕ : K ⊗_{F} \overline{F} ≅ Π_{i=1}^{n}\overline{F}

such that ϕ(α ⊗ 1) = (σ_{1}(α), …, σ_{n}(α)).

(g) If K/F is Galois find a smaller extension of F that can replace \overline{F} in part (f).

4. We defined a commutative ring to be Noetherian if every ideal is finitely generated. Section 15.1 of Dummit and Foote discusses some further properties of Noetherian.

(a) One of the more useful properties of Noetherian is that a commutative ring is Noetherian if and only if every submodule of a finitely generated R module is finitely generated. Prove this using any properties of Noetherian introduced in sections 12.1 and 15.1 of Dummit and Foote. (This is contained in exercises 6,7,8 on page 668 of Dummit and Foote.)

(b) Prove that a commutative ring is Noetherian if and only if every prime ideal of R is finitely generated (Exercise 11, page 668 of Dummit and Foote.)

(c) Let H be the ring of holomorphic functions in one variable z (i.e. power series in z that converge for all z). Prove that H is not Noetherian. Hint Remember that H contains infinite products, such as

sin(πz) = πz ∏_{k=1}^{∞}(1 − \frac{z^{2}}{k^{2}} ).

5. Let R ⊆ S be two integral domains. Let K be the field of fractions of S, and I = {x ∈ K|xS ⊆ R}.

(a) Prove that I ⊆ R and that I is an ideal of both R and S.

(b) Prove that if I ≠ 0 then R and S have the same field of fractions.

(c) Give an example of domains R and S (neither of which is a field) such that R and S have the same field of fractions, and I = 0.

(d) Let K be a field and t an indeterminate over K. Let R = K[t^{3}, t^{5}] and S = K[t]. Find I (giving a minimal set of generators for I as an R-ideal and as an S-ideal).

(e) Let k be a field and R = k[x_{1} … x_{n}]. Let I and J be two ideals in R generated by monomials. Prove that I ∩ J is also generated by monomials.

(f) Suppose that n = 2, I = (x, y)^{3} and J = (x). Find a minimial set of monomial generators for I ∩ J. In this question parts (e) and (f) are not particularly related to parts (a)-(d).

They are an attempt to equalize the effort required for each of the questions.

6. Let K be any field and let f ∈ K[X]. Then f is a “formal” expression f = a_{0} + a_{1}X + a_{2}X^{2} + ⋯ + a_{n}X^{n} with the familiar operations of addition and multiplication. To f ∈ K[X] we can associate a function from K to K by evaluation: a → f(a) = a_{0} +a_{1}a+a_{2}a^{2} +⋯ a_{n}a^{n}, where f(a) is calculated using the ring operations in the field K. If f ∈ K[X] it is important to remember that f itself is not a function, although one sometimes uses the same symbol f for the polynomial and the associated evaluation function, and might not distinguish clearly between the two concepts. In this question we wish to be more careful, and will write f̂ for the function f̂ : K → K defined by f̂(a) = a_{0} + a_{1}a + a_{2}a^{2} + ⋯ a_{n}a^{n}.

(a) Let R_{K} denote the set of all functions from K → K. Show that R_{K} is a ring under pointwise addition and multiplication (i.e. (f + g)(a) = f(a) + g(a) and (fg)(a) = f(a)g(a) for all a ∈ F and all f, g ∈ R_{F}.)

(b) Show that the map θ : K[X] → R_{K} defined by θ(f) = f̂ is a ring homomorphism.

(c) Show that if K is an infinite field then θ is one-to-one, but not onto.

(d) Now let p be a prime integer and K = ℤ/pℤ. How many elements are there in R_{ℤ/pℤ}?

(e) Show that if K = ℤ/pℤ then θ is onto but not one-to-one. What is the kernel of θ.