Cup product in Tate Cohomology Ring

Question

Let $G$ be a finite cyclic group of order $p$. The tate cohomology groups $\hat{H}^*(G, \mathbb{F}_p)$ are defined using a complete resolution of $\mathbb{F}$ as $\mathbb{F}_pG$-module.

There is a cup product on $\hat{H}^*(G,\mathbb{F_p})$, so I want to find the structure of this ring.

In this MO question

Examples of Tate cohomology rings

it is stated the integer case; however, I have not been able to find a reference to that statement.

For the integer case, there is an isomorphism $\hat{H^n}(G,\mathbb{Z}) \cong \hat{H^{n+d}}(G,\mathbb{Z})$ given by the cup product with some element $u$ of degree $d$ (the period). Does this also holds for the case $\mathbb{F}_p$?

Also, what techinque is used to compute the period of the tate cohomology and how to describe the ring?. For example, if $G$ is a cyclic group of order $2$, I think that $\hat{H}^*(G, \mathbb{F}_2) \cong \mathbb{F}_2[t,t^{-1}]$ because degreewise, $\hat{H^n}(G,\mathbb{F}_2) = \mathbb{F}_2$.

Answer 1

In Chapter XII.7 of the book

Cartan, Henri; Eilenberg, Samuel, Homological algebra, Princeton Mathematical Series. 19. Princeton, New Jersey: Princeton University Press xv, 390 p. (1956). ZBL0075.24305.

(starting on page 250) they calculate the cup products $\hat{H}^*(G;A)\otimes \hat{H}^*(G;A')\to \hat{H}^*(G;A\otimes A')$ explicitly, where $A,A'$ are $\mathbb{Z}G$-modules, using a map of resolutions. This gives a reference for the integer case, and probably it's not too hard to adapt thier argument to get the calculation for $\mathbb{F}_p$.