Math and stuff

Proposition Show that $\ev{xy, xz, yz}$ is a radical ideal. Solution Let $f \in k[x, y, z]$. Then $f$ is a finite linear combination of $x^ay^bz^c$ for some $a, b, c \geq 0$. If at least two of $a, b, c$ are positive, it can be expressed as a multiple...

Proposition Let $I, J, K \subset k[x_1, \cdots, x_n]$ be ideals. Prove the following: $IJ \subset K$ if and only if $I \subset K:J$. $(I:J):K = I:JK$. Solution 1 Suppose $IJ \subset K$. Let $x \in I$. Since $IJ \subset K$, $xJ \subset K$. Therefore, $x \in K:J$. Therefore, $I...

Proposition Determine whether the following polynomials lie in the following radicals. If the answer is yes, what is the smallest power of the polynomial that lies in the ideal? Is $x + y \in \sqrt{\ev{x^3, y^3, xy(x + y)}}$? Is $x^2 + 3xz \in \sqrt{\ev{x + z, x^2y, x -...

Proposition Let $J = \ev{xy, (x - y)x}$. Describe $V(J)$ and show that $\sqrt{J} = \ev{x}$. Solution from sympy import * from sympy.polys.orderings import monomial_key x, y, a = symbols('x y a') print(groebner([x*y, (x - y)*x], x, y, a, order='lex')) gives GroebnerBasis([x**2, x*y], x, y, a, domain='ZZ', order='lex'). Let $S...

Proposition Let $I_1, \cdots, I_n$ be ideals and $P$ a prime ideal containing $\cap_{i=1}^{n} I_i$. Then prove that $P \supset I_i$ for some $i$. Further, if $P = \cap_{i=1}^{n} I_i$, show that $P = I_i$ for some $i$. Solution Suppose $n = 2$. Then we want to show that $I_1...