That's a shame -- it's the way I would have done it, but the method I had in mind requires knowledge of finite fields.WannaBe22 said:The first way doesn't help me at all...
Maybe restating it more algebraically would help? For some other element a, You're looking to show the existence of a linear combination (coefficients in Z[x])t seems like the second way is the one we need to use... But I can't figure out what is the contradiction we get by assuming that there is an element k that exists in J but not in I...How does it help us?
The coefficients of linear combinations come from the ring. So, the element f(x) can be produced by f(x) * 1.WannaBe22 said:2. What is the unit element in Z[x]? If it's the same as the one in Z, So how come it "spans" the whole ring? I mean, if 1 is the unit element in Z[x], so how can we produce every element in Z[x] by multiplications of 1?
Abstract Algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. It is concerned with the properties and relationships between these structures and their operations.
Maximal ideals are special types of ideals in a ring that cannot be properly contained in any other ideal. In other words, they are the largest possible ideals in a given ring.
Maximal ideals play an important role in the structure and properties of rings. They are used to define quotient rings, which are important in understanding the structure of a ring and its ideals.
In the polynomial ring Z[x], maximal ideals are identified by the leading coefficient of the polynomial. Specifically, an ideal generated by a polynomial with a prime leading coefficient is a maximal ideal in Z[x].
Maximal ideals in Z[x] have various applications in number theory, algebraic geometry, and cryptography. They are also important in studying polynomial rings over other fields and their properties.