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Compute width of a $\mathbb Z_3$ extension over $\mathbb Z_3$

Could you please tell me how to solve this problem?
$$\mathbb{Z}_3[X]/(X^2+2X+1)$$ is a field extension of $\mathbb{Z}_3$?

A:

Define the ring $\mathbb Z_3[X]$ to be the set of polynomials with coefficients in $\mathbb Z_3$ and define the ideal $(X^2+2X+1)$ to be the set of polynomials with zero constant term and with coefficients in $\mathbb Z_3$ such that
$$\sum_{k\ge 0} a_kX^k \in(X^2+2X+1) \iff a_2=0$$
Now show that $X^2+2X+1$ is prime and that the polynomials with zero constant term are prime ideals.
Note that $(X^2+2X+1)$ is not maximal but the quotient by this ideal is an integral domain.

A:

Just an alternative,
The polynomial $X^2+2X+1$ has to remain irreducible. How to see that? By Eisenstein’s Criterion:
$$x^3+2x^2+2x+1=(x^3+1+x)^2-x(x^2+2x+1)$$
The prime polynomials

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Although sudden, internal auricular cardiac arrhythmias are commonly treated with supralethal electric shocks, lethal internal arrhythmias are rare. Nevertheless, lethal arrhythmias may cause sudden unexplained deaths. We present an unusual case of fatal ventricular arrhythmias in which a lethal leftanular laceration was caused by a supralethal shock applied by a standard defibrillator. With the increasing use of defibrillators in the public, and the fatal outcomes associated with their use, a forensic medicine viewpoint is offered regarding the post-sudden death blood stain analysis in the leftanular laceration in relation to the supralethal shock. The results show that the leftanular laceration in the cardiac area was caused by an accidental delivery of a biphasic shock.
//
// GCDropDownView.m
// CloverKit
//
// Created by Kevin Wojcik on 7/30/15.
//

#import “GCDropDownView.h”

@interface GCDropDownView()

@property (nonatomic, strong, readwrite) NSMutableArray *items;
@property (nonatomic, strong, readwrite) UIView *containerView;

@end

@implementation GCDropDownView

– (instancetype)initWithFrame:(CGRect)frame
{
self = [super initWithFrame:frame];
if (self) {
_items = [NSMutableArray array];
_containerView = [[UIView alloc] init];