Flashcards
Galois Chapter 1
Cards in this deck (31)
- 1What is an integral domain?A non commutative ring with unity which has no zero divisors. I.e. if \( ab=0_{R} \) then at least one of \( a \) or \(b\) is zero.
- 2What is a field?A non trivial commuting ring with unity in which every non-zero element has a multiplicative inverse.
- 3Is every field an integral domain?Yes.
- 4What is the characteristic of a field?The smallest positive integer such that \( n.1_{R}=0 \). It is denoted \( \chi(R) \).
- 5Define the terms indeterminate, degree, leading coefficient, and monic.Indeterminate is not a quantity, it is used to describe the polynomial.
The degree of a polynomial is the highest order term.
The leading coefficient is the coefficient of the highest order term.
A polynomial is monic if it's leading coefficient is one. - 6What is the field of rational expressions?It is defined over an integral domain, and is the set of all quotients of polynomials in \(D[t]\).
- 7Give the definition of \(\delta(f)\) and therefore the discriminant.\[\delta(f) = \prod_{1 \leq i < j \leq n}(\alpha_{i}-\alpha{j}).\] The discriminant is \( \Delta(f) = \delta(f)^{2} \).
- 8What condition must hold for there to be a repeated root of \(f\)?We must have a shared root \(\alpha\) between \(f\) and \(Df\).
- 9State the rational root theorem.Suppose \(f\) has a rational zero \(\frac{p}{q}\). Then \(p\) divides \(a_{0}\) and \(q\) divides \(a_{n}\).
- 10Write the monic polynomial of degree \(n\) with zeros \(\alpha_{1},...,\alpha_{n} \) in terms of the elementary symmetric functions.\[f=t^n - s_{1}t^{n-1} + s_{2}t^{n-2} - ... +(-1)^{n}s_{n}. \]
- 11What is a primitive root of unity?A root of unity that generates all the others.
- 12What is the Tschirnhaus Transformation and what does it do?Substitute \(x=y-\frac{a_{n-1}}{n}\). This removes the second highest order term from the polynomial.
- 13What is the Vieta substitution and how does it help?Substitute \(y=z-\frac{p}{z}\). This gives us a quadratic in \(z^3\) that we can use to solve the original cubic. Note it does not matter which of the solutions to the quadratic we use.
- 14What is the determinant of a cubic (assuming that we have already eliminated the quadratic term)?\(\Delta(f) = -4p^3-27q^2. \)
- 15Define 'properly reducible'.A polynomial \(f\in D[t]\), of degree \(\geq 1 \) is properly reducible if \(f\) is the product of two polynomials also in \(D[t]\). Note that over a field, reducible and properly reducible are the same.
- 16When is a polynomial of degree two or three reducible?When the polynomial has at least one zero in \(K\).
- 17What is a ring homomorphism?A map between rings (or fields) such that \(\phi(r_1+r_2)=\phi(r_1)+\phi(r_2)\) and \( \phi(r_{1}r_{2})=\phi(r_1)\phi(r_2). \)
- 18What is the natural homomorphism?The map \( v_n(a) = a \) mod \( n \). It maps from \( \mathbb{Z} \) into \(\mathbb{Z}_n\).
- 19Define the content of a polynomial, explain what it means for f to be primitive.The content \(c_f\) is the greatest common divisor of the coefficients of \(f\). A polynomial is primitive if its content is 1.
- 20What is Gauss's Lemma?Let \(f\in\mathbb{Z}[t]\). Then \(f\) is reducible over \(\mathbb{Q}\) if and only if it is properly reducible over \(\mathbb{Z}\).
- 21What is the localization principle?vLet \(f\in\mathbb{Z}[t]\) and let \(p\) be a prime which does not divide the leading coefficient. If \(v_{p}(f) \) is reducible over \(\mathbb{F}_p\) then \(f\) is reducible over \(\mathbb{Q}\).
- 22What is Eisenstiens Irreducibility Criterion (EIC)?If there exists a prime \(p\) which divides each of \(a_0,...a_{n-1}\) but not \(a_n\), and \(p^2\) does not divide \(a_0\), then f is irreducible over \(\mathbb{Q}\).
- 23State the "reverse coefficient" proposition.A polynomial \(a_0 + a_{1}t+...+a_{n}t^n\) is reducible over \(\mathbb{Q}\) if and only if \(a_{0}t^n+a_{1}t+...+a_n\) is also reducible over \(\mathbb{Q}\).
- 24Define the cyclotomic polynomials.The n'th cyclotomic polynomial is given by \[\Phi_n = \prod_{i=1}^{n} (t-\epsilon_i)\] where \(\epsilon_1,...,\epsilon_n\) are all the primitive n'th roots of unity.
- 25Is the cyclotomic polynomial irreducible over \(\mathbb{Q}\)?For each prime \(p\), the cyclotomic polynomial \[\Phi_p = \prod_{r=0}^{p-1}(t^r)\] is irreducible over \(\mathbb{Q}\).
- 26What is a an algebraic/transcendental element?An element \(\alpha \in L\) is algebraic over \(K\) if there exists \(f\inK[t]\) such that \(f(\alpha)=0\). Otherwise \(\alpha\) is transcendental over \(K\).
- 27Let \(\alpha\) be algebraic over a field \(K\), and let \(\mu\in K[t]\), of lowest degree among those that have \(\alpha\) as a zero. Then...\(\mu\) is irreducible over \(K\), and unique. If \(f(\alpha)=0\) for some \(f\in K[t]\), then \(\mu | f\).
- 28What is the definition of the minimal polynomial of \(\alpha\) over \(K\)?The lowest degree monic polynomial \(\mu \in K[t]\) such that \(\mu(\alpha)=0\). We say that \(\alpha\) is algebraic of degree \(\partial\mu\) over \(K\).
- 29What are the conjugates of \(\alpha\) over \(K\)?The other roots of the minimal polynomial of \(\alpha\) over \(K\).
- 30What does it mean for a polynomial to be separable?\(f\in K[t] \) is separable over \(K\) if no irreducible factor of \(f\in K[t]\) has a repeated root. Otherwise it is inseparable
- 31Let \(K\) be a field of characteristic zero, or a finite field. Then...Every irreducible polynomial in \(K[t]\) is separable over \(K\).
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- Mathematics Flashcards on Galois Chapter 1, created by James Woodcock on 25/03/2017.
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- James Woodcock
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